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A local field is a complete discretely valued field with finite residue field—a notion that unifies the real and complex numbers, the p-adic numbers, and finite extensions thereof. Local Fields is a course in algebraic number theory focused on understanding the structure and arithmetic of these fields and their extensions. Rather than studying global number fields like the rationals or algebraic integers, we zoom in on the local picture: the completion at a single prime. This localization strategy is powerful because many global questions reduce to local ones, and local fields offer a more concrete setting where explicit computations and classification theorems become possible. The course builds systematically from foundations to deep structural results. We begin with basic properties of complete valued fields and the topology they carry, then introduce the general framework of valued fields and their extensions. The core chapters focus on discretely valued fields—where valuations take values in $\mathbb{Z}$—developing ramification theory, which measures how primes split and how valuations extend to larger fields. This theory culminates in local class field theory, which describes abelian extensions of local fields via the local reciprocity map, and Lubin-Tate theory, which provides explicit constructions of these extensions using formal groups. Throughout, the interplay between algebraic structure (field extensions, Galois groups) and analytic/topological properties (completeness, valuation theory) is central. The p-adic numbers—the completion of the rationals at each prime p—serve as the canonical example and motivation for the entire course. By the end, you will understand how local fields encode arithmetic information, how to compute with them, and why they are indispensable in modern number theory. These notes accompany the Cambridge Part III course on Local Fields. The course develops the theory of $p$-adic numbers and their abstractions — local fields — from first principles, weaving together algebra, analysis, and topology. The central objects are non-archimedean valued fields, and the interplay between their metric and algebraic structure is what makes the subject both subtle and powerful. ## Why Local Fields? [motivation] ### The problem of solving polynomial equations Many fundamental problems in number theory begin with a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z}[x_1, \ldots, x_n]$ and the question: does $f$ have a solution $a \in \mathbb{Z}^n$? Two classical strategies offer partial answers. The first strategy is geometric: view $f$ as a real polynomial, analyse its real roots, and ask whether any are integers. This works when $f$ has no real solutions at all — in that case the answer over $\mathbb{Z}$ is immediately no. The second strategy is arithmetic: reduce $f$ modulo a prime $p$. If $f$ has no solution mod $p$, it has no integer solution either. But this is rarely decisive on its own. The equation may have solutions mod every prime $p$, yet fail to have an integer solution — this phenomenon, the failure of the Hasse principle, is exactly what makes the problem hard. To sharpen the modular approach, one studies solutions modulo $p^2$, $p^3$, and all higher powers simultaneously. ### Packaging congruence conditions into a single field The $p$-adic numbers $\mathbb{Q}_p$, invented by Hensel in the late 19th century, achieve precisely this packaging. An element of $\mathbb{Q}_p$ is a coherent system of congruence conditions modulo every power of $p$. Concretely, every $a \in \mathbb{Z}_p$ has a unique expansion \begin{align*} a = \sum_{i=0}^{\infty} a_i p^i, \qquad a_i \in \{0, 1, \ldots, p-1\}, \end{align*} which one can think of as an integer in base $p$ allowed to be infinitely long to the left. The ring $\mathbb{Z}_p$ knows about $\mathbb{Z}/p^n\mathbb{Z}$ for every $n$ simultaneously: the natural map $\mathbb{Z}_p \to \mathbb{Z}/p^n\mathbb{Z}$ is surjective with kernel $p^n\mathbb{Z}_p$, and $\mathbb{Z}/p^n\mathbb{Z} \cong \mathbb{Z}_p/p^n\mathbb{Z}_p$. The field $\mathbb{Q}_p$ is the fraction field of $\mathbb{Z}_p$, obtained by completing $\mathbb{Q}$ under the $p$-adic absolute value $|\cdot|_p$. The $p$-adic absolute value reverses the usual hierarchy of size: $p^n \to 0$ as $n \to \infty$, so numbers divisible by high powers of $p$ are considered small. This makes $\mathbb{Q}_p$ a non-archimedean field — the triangle inequality is replaced by the much stronger ultrametric inequality $|x + y|_p \leq \max(|x|_p, |y|_p)$. ### From $\mathbb{Q}_p$ to local fields The concept of a local field is an abstraction of $\mathbb{Q}_p$. Rather than working with the specific prime $p$, one studies fields carrying a non-archimedean absolute value (equivalently, a discrete valuation) that are complete with respect to the induced metric and have finite residue field. The theory develops in two parallel streams: the characteristic-zero fields (finite extensions of $\mathbb{Q}_p$) and the characteristic-$p$ fields (formal Laurent series $\mathbb{F}_q((t))$ over finite fields). Both are encompassed by a single axiomatic framework. In this course, we focus not on the number-theoretic applications but on the intrinsic structure of local fields: their topology, their algebraic extensions, ramification theory, and the rich interaction between the valuation and the Galois group of a finite extension. [/motivation] ## Course Overview and Prerequisites The course assumes familiarity with basic algebra (groups, rings, fields), including Galois theory, and with the rudiments of point-set topology and metric spaces. Prior exposure to number fields is helpful but not required. The main topics covered are: - The $p$-adic numbers: construction via completion, structure of $\mathbb{Z}_p$ and $\mathbb{Q}_p$, $p$-adic expansions. - Valued fields and local fields: absolute values, non-archimedean valuations, valuation rings, the general framework of local fields. - Finite extensions: extension of valuations, ramification theory (inertia group, ramification filtration, tamely and wildly ramified extensions). - Polynomial equations: Hensel's Lemma and its applications, Newton polygons as a tool for factoring polynomials over local fields. - Continuous functions on $\mathbb{Z}_p$: Mahler's theorem characterising continuous functions $\mathbb{Z}_p \to \mathbb{Q}_p$ via Mahler coefficients. - Local class field theory (time permitting): the reciprocity map, the structure of abelian extensions of a local field. **Convention.** Throughout the course, all rings are commutative with unity unless explicitly stated otherwise. ## The $p$-Adic Absolute Value How do we make the idea of "$p$-divisibility" into a genuine metric on $\mathbb{Q}$? The answer is to measure the size of a rational number by how many times $p$ divides it — with higher divisibility meaning smaller size. Every nonzero rational $x \in \mathbb{Q}$ can be written uniquely as \begin{align*} x = p^n \frac{a}{b}, \end{align*} where $n \in \mathbb{Z}$, $a, b \in \mathbb{Z}$ with $b > 0$, and $\gcd(a, p) = \gcd(b, p) = 1$. The exponent $n$ is the $p$-adic valuation of $x$, written $v_p(x) = n$; to compute it, factor the numerator and denominator of $x$ completely and set $v_p(x) = (\text{power of } p \text{ in the numerator}) - (\text{power of } p \text{ in the denominator})$. [definition: P-Adic Absolute Value] Let $p$ be a prime. The **$p$-adic absolute value** on $\mathbb{Q}$ is the function $|\cdot|_p: \mathbb{Q} \to \mathbb{R}_{\geq 0}$ defined by \begin{align*} |x|_p = \begin{cases} 0 & x = 0, \\ p^{-n} & x = p^n \tfrac{a}{b} \text{ with } \gcd(a,p) = \gcd(b,p) = 1. \end{cases} \end{align*} [/definition] The effect of this definition is that $|p^n|_p = p^{-n}$: elements highly divisible by $p$ are small, while elements with $p$ in the denominator are large. For instance, $|p^3|_p = p^{-3}$ is tiny, while $|p^{-2}|_p = p^2$ is large. [example: P-Adic Sizes of Integers] Let $p = 5$. Consider the integers $1, 5, 25, 125$. Their $5$-adic absolute values are \begin{align*} |1|_5 = 1, \qquad |5|_5 = 5^{-1}, \qquad |25|_5 = 5^{-2}, \qquad |125|_5 = 5^{-3}. \end{align*} So the sequence $1, 5, 25, 125, \ldots$ converges to $0$ in $\mathbb{Q}_5$. In fact, the geometric series $\sum_{k=0}^{\infty} 5^k$ converges in $\mathbb{Q}_5$, because its terms tend to zero. Its sum is $\frac{1}{1 - 5} = -\frac{1}{4}$, since $|5^k|_5 = 5^{-k} \to 0$. This already illustrates one of the key convergence phenomena: in a non-archimedean field, a series converges if and only if its terms tend to zero. [/example] [example: P-Adic Absolute Value of a Rational] Let $x = \frac{63}{550}$. We compute $|x|_p$ at several primes by factoring numerator and denominator: \begin{align*} 63 &= 2^0 \cdot 3^2 \cdot 5^0 \cdot 7, \\ 550 &= 2 \cdot 5^2 \cdot 11. \end{align*} Thus $v_p\!\left(\tfrac{63}{550}\right) = v_p(63) - v_p(550)$ gives: \begin{align*} v_2\!\left(\tfrac{63}{550}\right) &= 0 - 1 = -1, &|x|_2 &= 2^{1} = 2, \\ v_3\!\left(\tfrac{63}{550}\right) &= 2 - 0 = 2, &|x|_3 &= 3^{-2} = \tfrac{1}{9}, \\ v_5\!\left(\tfrac{63}{550}\right) &= 0 - 2 = -2, &|x|_5 &= 5^{2} = 25, \\ v_7\!\left(\tfrac{63}{550}\right) &= 1 - 0 = 1, &|x|_7 &= 7^{-1} = \tfrac{1}{7}, \\ v_{11}\!\left(\tfrac{63}{550}\right) &= 0 - 1 = -1, &|x|_{11} &= 11^{1} = 11. \end{align*} The same rational number has a completely different size depending on which prime we measure with. At $p = 3$ and $p = 7$ it is small (numerator divisible), while at $p = 5$ and $p = 11$ it is large (denominator divisible). [/example] The $p$-adic absolute value satisfies the strong triangle inequality $|x + y|_p \leq \max(|x|_p, |y|_p)$, which is the defining property of a non-archimedean absolute value. This will be established rigorously in Chapter 1, together with the full theory of non-archimedean valued fields.  [remark: Why Non-Archimedean?] The name "non-archimedean" refers to the failure of the Archimedean property: in $\mathbb{Q}_p$, the integers $1, 2, 3, \ldots$ are not unbounded — they all satisfy $|n|_p \leq 1$, since no integer has $p$ in its denominator. This is the opposite of the situation in $\mathbb{R}$, where $n \to \infty$. The Archimedean property characterises $\mathbb{R}$ and $\mathbb{C}$ among all completions of $\mathbb{Q}$ (a classical theorem of Ostrowski), and the $p$-adic absolute values account for all the non-archimedean completions. [/remark] # Introduction This chapter lays the algebraic and analytic foundations for the entire course. We introduce absolute values on fields and study the striking distinction between archimedean and non-archimedean valuations. The bulk of the chapter develops the non-archimedean theory: the valuation ring, the $I$-adic topology, completions via inverse limits, and finally the $p$-adic numbers $\mathbb{Q}_p$ as the canonical example that motivates everything else. ## Absolute Values and Valued Fields To measure the "size" of elements in a field $K$, we need a function that satisfies the same axioms as the familiar absolute value on $\mathbb{R}$. [definition: Absolute Value] Let $K$ be a field. An **absolute value** on $K$ is a function $|\cdot|: K \to \mathbb{R}_{\geq 0}$ satisfying: 1. $|x| = 0$ if and only if $x = 0$; 2. $|xy| = |x||y|$ for all $x, y \in K$; 3. $|x + y| \leq |x| + |y|$ (the triangle inequality). A **valued field** is a field equipped with an absolute value. [/definition] The definition axiomatises exactly what is needed to do analysis on a field: a notion of size that interacts well with both addition and multiplication. The triangle inequality in particular is what makes Cauchy sequences and convergence well-behaved. [example: The $p$-Adic Absolute Value on $\mathbb{Q}$] Fix a prime $p$. Every non-zero rational $x \in \mathbb{Q}^\times$ can be written uniquely as $x = p^n a/b$ with $n \in \mathbb{Z}$, $a, b \in \mathbb{Z}$, and $\gcd(a, p) = \gcd(b, p) = 1$. Set \begin{align*} |x|_p = p^{-n}, \qquad |0|_p = 0. \end{align*} This is an absolute value: non-degeneracy is immediate, multiplicativity follows from $v_p(xy) = v_p(x) + v_p(y)$, and the strong triangle inequality $|x + y|_p \leq \max(|x|_p, |y|_p)$ holds because the $p$-adic valuation of a sum is at least the minimum of the two valuations. We will verify the strong triangle inequality in full below; the key point here is that $|\cdot|_p$ assigns small absolute value to elements highly divisible by $p$ — the deeper $p$ divides $x$, the "smaller" $x$ is. For example, $|p^{100}|_p = p^{-100}$, while $|7|_p = 1$ for any prime $p \neq 7$. [/example] The $p$-adic absolute value is the most important example for this course, but it helps to contrast it with the opposite extreme: an absolute value that carries no analytic information whatsoever. [example: The Discrete Absolute Value] Any field $K$ admits the **discrete absolute value**, defined by \begin{align*} |x| = \begin{cases} 1 & x \neq 0 \\ 0 & x = 0. \end{cases} \end{align*} This does satisfy the axioms, but it generates the discrete topology on $K$ and carries no interesting analytic information. For this reason, we assume throughout the course that all absolute values are non-degenerate in the sense of generating a non-discrete topology. [/example] Every absolute value defines a metric $d(x, y) = |x - y|$ on $K$, turning $K$ into a metric space. The basic geometric intuition from $\mathbb{R}$ carries over: convergence, Cauchy sequences, and completions all make sense in this generality. One basic consequence is the **reverse triangle inequality**: [quotetheorem:2300] [citeproof:2300] The reverse triangle inequality is the analytic bedrock of valued fields: it ensures that the map $x \mapsto |x|$ is continuous, so limits interact well with the absolute value. Without it one could not even make sense of completions. Two absolute values on the same field are **equivalent** if they define the same topology. Since the topology determines convergence, equivalent absolute values are analytically interchangeable. The following proposition gives a clean algebraic characterisation. [definition: Equivalence of Absolute Values] Let $K$ be a field and let $|\cdot|$, $|\cdot|'$ be absolute values on $K$. They are **equivalent** if they induce the same topology on $K$. [/definition] The question of when two absolute values carry the same analytic content reduces to a clean algebraic criterion. The following theorem shows that the topology — which governs convergence, continuity, and completions — is the only invariant that matters. [quotetheorem:2301] [citeproof:2301] Condition (iii) says that equivalent absolute values are simply positive powers of each other. In particular, every equivalence class of absolute values contains a canonical representative. The characterisation via condition (ii) is especially useful: it says that what matters is not the precise value $|x|$ assigns, but only which elements are "small" (have absolute value $< 1$). This open unit disk $\{|x| < 1\}$ is a canonical invariant of the equivalence class. [remark: Completions Are Equivalence-Invariant] Equivalent absolute values on $K$ induce the same completion $\hat{K}$. The completion $\hat{K}$ is itself a valued field, with a unique absolute value extending $|\cdot|$ that makes $K \hookrightarrow \hat{K}$ an isometric embedding. We will construct completions explicitly in the section on $I$-adic topology. [/remark] ## Non-Archimedean Absolute Values The classical absolute values on $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ satisfy the Archimedean property: for any $x \neq 0$, the sequence $|x|, |2x|, |3x|, \ldots$ is unbounded. This is equivalent to the triangle inequality being "sharp" in the sense that $|n \cdot 1|$ grows without bound. Local fields are characterised by the opposite behaviour. [definition: Non-Archimedean Absolute Value] An absolute value $|\cdot|$ on a field $K$ is **non-archimedean** if it satisfies the **strong triangle inequality**: \begin{align*} |x + y| \leq \max(|x|, |y|) \quad \text{for all } x, y \in K. \end{align*} An absolute value that is not non-archimedean is called **archimedean**. A metric $d$ satisfying $d(x,z) \leq \max(d(x,y), d(y,z))$ is called an **ultrametric**. [/definition] The strong triangle inequality is genuinely stronger than the ordinary triangle inequality: if $|x + y| \leq \max(|x|, |y|)$, then in particular $|x + y| \leq |x| + |y|$. For the rest of this course, unless otherwise stated, all absolute values are assumed to be non-archimedean. The first striking consequence of the strong triangle inequality is that geometry in a non-archimedean field looks nothing like geometry in $\mathbb{R}$.  [quotetheorem:2302] [citeproof:2302] [remark: Ultrametric Geometry] The theorem says that every element of a closed ball is equally entitled to be called its center. Two closed balls of the same radius are either disjoint or identical — they can never "partially overlap." This is the defining strangeness of ultrametric geometry. [/remark] A corollary is that, in the non-archimedean world, closed balls are open sets. This is counterintuitive from the perspective of $\mathbb{R}$, where the open and closed balls of radius $r$ around $x$ are genuinely different sets ($]x-r, x+r[$ vs $[x-r, x+r]$). The non-archimedean strong triangle inequality obliterates this distinction. [quotetheorem:2303] [citeproof:2303] This result has no analogue in archimedean analysis: in $\mathbb{R}$, the closed ball $[x - r, x + r]$ is never an open set. The underlying reason is that the non-archimedean strong triangle inequality forces the topology to be totally disconnected — the connected components of $K$ are singletons. Every point is isolated in the sense that it admits a basis of neighbourhoods that are simultaneously open and closed (clopen). This total disconnectedness is a recurring theme: it is what makes the valuation ring $\mathcal{O}_K$ simultaneously open and closed in $K$, and it will be essential when we later prove that $\mathcal{O}_K$ is compact (in the complete, discretely-valued case). Non-archimedean fields are also much simpler than $\mathbb{R}$ when it comes to computing the absolute value of a sum. [quotetheorem:2304] [citeproof:2304] The informal principle here is that "in a non-archimedean field, all triangles are isoceles." If two sides of a triangle have different lengths, the third must equal the longer one. This principle is remarkably useful in practice: to compute $|x + y|_p$, one only needs to know which of $|x|_p$ or $|y|_p$ is larger — the answer is immediate. The equality case $|x| = |y|$ is the only genuinely subtle situation. ## Convergence in Non-Archimedean Fields One of the most useful consequences of the strong triangle inequality is a dramatic simplification of convergence criteria. In ordinary analysis, showing that a series converges requires careful estimates on the tail. In a non-archimedean complete field, one only needs to check that the terms go to zero. [quotetheorem:2305] [citeproof:2305] This should be compared with the situation over $\mathbb{R}$: the harmonic series $\sum 1/n$ has terms tending to $0$ but diverges. In a non-archimedean complete field, the terms tending to $0$ is sufficient. ## The Valuation Ring The rich algebraic structure of non-archimedean valued fields is concentrated in a canonical subring, the valuation ring. [definition: Valuation Ring] Let $(K, |\cdot|)$ be a non-archimedean valued field. The **valuation ring** of $K$ is \begin{align*} \mathcal{O}_K = \{x \in K : |x| \leq 1\}. \end{align*} [/definition] The valuation ring is an open subring, because (as we proved) closed balls are open sets. For ordinary absolute values on $\mathbb{R}$, the analogous set $\{x : |x| \leq 1\} = [-1, 1]$ is not a ring: $1 + 1 = 2 \notin [-1, 1]$. The non-archimedean condition is essential — it is precisely what allows the set of "bounded" elements to be closed under addition. [quotetheorem:2306] [citeproof:2306] The theorem identifies the three layers of $\mathcal{O}_K$: the whole ring, the maximal ideal $\mathfrak{m}_K = \{x : |x| < 1\}$ (the unique proper open ideal), and the unit group $\mathcal{O}_K^\times = \{x : |x| = 1\}$. These three pieces completely stratify $\mathcal{O}_K$, a fact that will recur constantly when we study the structure of extensions. ## Integral Elements and Integral Closure When we extend a non-archimedean valued field $K$ to a finite extension $L/K$, the valuation ring $\mathcal{O}_K$ sits inside $L$. A fundamental question is: which elements of $L$ should be regarded as "integers" — that is, as analogues of $\mathcal{O}_K$ in the larger field? The algebraic answer is integrality: an element that satisfies a monic polynomial equation over $\mathcal{O}_K$. Without this notion, we would have no way to identify the ring of integers of a finite extension, which is the central object of study for the rest of the course. To handle ring extensions and later develop the theory of finite extensions of local fields, we need the notion of integrality. [definition: Integral Element] Let $R \subseteq S$ be a ring extension. An element $s \in S$ is **integral over $R$** if there exists a monic polynomial $f \in R[x]$ with $f(s) = 0$. [/definition] [example: Algebraic Integers] In the extension $\mathbb{Z} \subseteq \mathbb{C}$, an element $z \in \mathbb{C}$ is integral over $\mathbb{Z}$ precisely when it is an algebraic integer. For instance, $\sqrt{2}$ satisfies $x^2 - 2 = 0$ and is integral. On the other hand, $1/\sqrt{2}$ is not integral over $\mathbb{Z}$: any monic polynomial $x^n + a_{n-1}x^{n-1} + \cdots + a_0$ with integer coefficients evaluated at $1/\sqrt{2}$ cannot vanish (since by the rational root theorem, rational roots of monic integer polynomials must be integers, and $1/\sqrt{2}$ is irrational; a more careful argument shows non-integrality). [/example] The following theorem characterises integrality in terms of finite generation, which is the key tool for proving that sums and products of integral elements are integral. [quotetheorem:2307] [citeproof:2307] The adjugate matrix argument here deserves a word. The key move is to encode the equation $s \cdot t_i = \sum_j a_{ij} t_j$ in matrix form $(sI - A)t = 0$, and then multiply through by the adjugate $(sI - A)^{\text{adj}}$ to extract $\det(sI - A) = 0$. This is a linear-algebraic trick that converts a system of finitely many relations into a single polynomial equation. It is the engine behind almost all finiteness arguments in commutative algebra, and it will appear again when we study the integral closure of $\mathcal{O}_K$ in finite extensions. [quotetheorem:2308] [citeproof:2308] This is the cleanest proof of the ring property: rather than struggling with explicit polynomial equations for $s_1 + s_2$ and $s_1 s_2$, one works in the finitely generated module $R[s_1, s_2]$ and uses the general characterisation. The key insight is that integrality is really a statement about finite generation of modules, not about polynomials per se. [definition: Integrally Closed] A ring $R$ in an extension $R \subseteq S$ is **integrally closed** in $S$ if $\tilde{R} = R$, i.e., every element of $S$ integral over $R$ already belongs to $R$. [/definition] Valuation rings $\mathcal{O}_K$ are always integrally closed in $K$: if $x \in K$ satisfies $x^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0$ with all $a_i \in \mathcal{O}_K$, then $|x|^n = |a_{n-1}x^{n-1} + \cdots + a_0| \leq \max(|x|^{n-1}, 1)$, which forces $|x| \leq 1$. This integral-closure property is a fundamental reason why valuation rings behave so nicely in algebraic number theory. ## Topological Rings and the $I$-Adic Topology The definition of a topological ring packages two demands: the algebra (the ring structure) must be compatible with the geometry (the topology). Over $\mathbb{Q}$ with the standard absolute value, addition and multiplication are continuous — this is so automatic we rarely mention it. But consider the ring $\mathbb{Z}$ with an exotic topology: not every topology makes $\mathbb{Z}$ a topological ring. The $I$-adic topology is a canonical choice that always works, and it is the right notion for studying completions. The valuation ring $\mathcal{O}_K$ inherits from $K$ a topology in which the ring operations are continuous. This is the prototypical example of a topological ring. [definition: Topological Ring] A **topological ring** is a ring $R$ with a topology such that addition $R \times R \to R$ and multiplication $R \times R \to R$ are both continuous (with $R \times R$ given the product topology). [/definition] A valued field is automatically a topological ring, since addition and multiplication are continuous with respect to the metric topology. More generally, given any ideal $I$ of a ring $R$, we can construct a canonical ring topology by declaring that elements of $I^n$ are "small of order $n$." The intuition: in the $x$-adic topology on $R$, where $I = xR$, we want the power series $a_0 + a_1 x + a_2 x^2 + \cdots$ to converge. The partial sums $\sum_{i=0}^{n-1} a_i x^i$ approximate the limit modulo $I^n$, so convergence in this topology means convergence modulo every $I^n$. [definition: $I$-Adic Topology] Let $R$ be a ring and $I \subseteq R$ an ideal. A subset $U \subseteq R$ is **$I$-adically open** if for every $x \in U$, there exists $n \geq 1$ such that $x + I^n \subseteq U$. The collection of all $I$-adically open sets forms the **$I$-adic topology** on $R$. [/definition] Having defined the topology, the first thing to check is that the ring operations remain continuous — without this, the algebraic and topological structures would be disconnected, and the resulting theory would be unworkable. The next theorem confirms that the $I$-adic topology always produces a topological ring. [quotetheorem:2309] [citeproof:2309] When $I = xR$ is the principal ideal generated by $x$, we call this the **$x$-adic topology**. The $I$-adic topology on $R$ is the one in which the ideals $I^n$ serve as "neighbourhoods of zero." An element $x \in R$ is "close to zero" if it belongs to a high power of $I$. This matches the intuition for $p$-adic closeness: $x$ is close to $y$ in the $p$-adic sense if $p^n \mid x - y$ for large $n$, i.e., $x - y \in p^n \mathbb{Z}$. ## Completions and the Inverse Limit To formalise the notion of $I$-adic completeness, we need the categorical construction of the inverse limit, which packages together the successive approximations $R/I^n$ into a single ring. [definition: Inverse Limit] Let $R_1, R_2, \ldots$ be topological rings with continuous ring homomorphisms $f_n: R_{n+1} \to R_n$. The **inverse limit** (or **projective limit**) is the ring \begin{align*} \varprojlim R_n = \bigl\{(x_n) \in \prod_{n \geq 1} R_n : f_n(x_{n+1}) = x_n \text{ for all } n \bigr\}, \end{align*} with coordinate-wise addition and multiplication, equipped with the subspace topology inherited from the product topology on $\prod_n R_n$. This is the **inverse limit topology**. [/definition] The inverse limit inherits its ring structure coordinate-wise from the factors $R_n$, and the subspace topology from the product guarantees that the projections $\varprojlim R_n \to R_m$ are continuous. The key verification is that addition and multiplication on $\varprojlim R_n$ are themselves continuous — this is the content of the following theorem. [quotetheorem:2310] [citeproof:2310] The inverse limit satisfies a universal property: a continuous ring homomorphism $g: S \to \varprojlim R_n$ is the same data as a compatible family of continuous ring homomorphisms $g_n: S \to R_n$ satisfying $f_n \circ g_{n+1} = g_n$ for all $n$. This universal property is what makes the inverse limit the right categorical tool for completions: it says that maps into the completed object are exactly compatible systems of maps into the approximations $R_n$. An element of $\varprojlim R_n$ is therefore not some mysterious limit object — it is simply a coherent set of instructions, one instruction per level. The conceptual power of inverse limits is that they transform an infinite process (passing to successively finer approximations) into a single algebraic object. Instead of thinking about sequences of partial sums, one thinks about the compatible family of their residues modulo $I^n$. This algebraic packaging makes it much easier to verify continuity and to study ring homomorphisms into the completion. We now apply this to define completions. Given a ring $R$ and ideal $I$, the quotients $R/I^n$ form an inverse system under the natural surjections $R/I^{n+1} \to R/I^n$. [definition: $I$-Adic Completion] Let $R$ be a ring and $I \subseteq R$ an ideal. The **$I$-adic completion** of $R$ is the topological ring \begin{align*} \hat{R} = \varprojlim R/I^n, \end{align*} where each $R/I^n$ carries the discrete topology and the transition maps are the natural quotient maps $R/I^{n+1} \to R/I^n$. There is a canonical continuous ring homomorphism \begin{align*} \nu: R &\to \hat{R}, \quad r \mapsto (r \bmod I^n)_{n \geq 1}, \end{align*} where $R$ is given the $I$-adic topology. We say $R$ is **$I$-adically complete** if $\nu$ is a bijection (equivalently, a homeomorphism). [/definition] [remark: What Completeness Means] An element of $\hat{R} = \varprojlim R/I^n$ is a coherent sequence of approximations: a choice of $x_n \in R/I^n$ for each $n$ with $x_n \equiv x_{n+1} \pmod{I^n}$. The ring $R$ is $I$-adically complete when every such coherent sequence is actually "the sequence of partial reductions of some element $r \in R$," i.e., $x_n = r \bmod I^n$ for all $n$. This is exactly the $I$-adic analogue of Cauchy completeness. [/remark] ## The $p$-Adic Numbers For the rest of this course, $p$ denotes a fixed prime. The $p$-adic numbers are the fundamental example of a non-archimedean locally compact field, and the entire course is built around understanding their structure and that of their finite extensions. The construction is entirely elementary — we are just completing $\mathbb{Q}$ with respect to the $p$-adic absolute value — but the resulting field $\mathbb{Q}_p$ has a rich and non-obvious structure that we will spend the rest of the course unpacking. Every non-zero rational number $x \in \mathbb{Q}^\times$ can be written uniquely as \begin{align*} x = p^n \frac{a}{b}, \end{align*} where $n \in \mathbb{Z}$ and $a, b \in \mathbb{Z}$ with $b > 0$ and $\gcd(a, p) = \gcd(b, p) = 1$. [definition: $p$-Adic Absolute Value] The **$p$-adic absolute value** on $\mathbb{Q}$ is the function $|\cdot|_p : \mathbb{Q} \to \mathbb{R}_{\geq 0}$ defined by \begin{align*} |x|_p = \begin{cases} 0 & x = 0 \\ p^{-n} & x = p^n \frac{a}{b} \text{ as above.} \end{cases} \end{align*} [/definition] The $p$-adic absolute value assigns small values to integers highly divisible by $p$. For $x \in \mathbb{Z}$, we have $|x|_p = p^{-n}$ iff $p^n \mid x$ but $p^{n+1} \nmid x$ (written $p^n \| x$). This turns the intuition "divisible by high powers of $p$ means close to $0$" into a precise metric. [quotetheorem:2311] [citeproof:2311] The fact that $|\cdot|_p$ is non-archimedean — and not merely a metric — is not a coincidence. It reflects the fundamental arithmetic of divisibility: $p$ divides a sum only if $p$ divides both summands (or at least one to a high power), so the worst-case divisibility of $x + y$ is at least the minimum of the divisibilities of $x$ and $y$. The non-archimedean property is the analytic shadow of this algebraic fact. [definition: $p$-Adic Numbers and $p$-Adic Integers] The **$p$-adic numbers** $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ with respect to $|\cdot|_p$. The **$p$-adic integers** are the valuation ring \begin{align*} \mathbb{Z}_p = \{x \in \mathbb{Q}_p : |x|_p \leq 1\}. \end{align*} [/definition] The definition of $\mathbb{Q}_p$ as a completion is entirely analogous to the construction of $\mathbb{R}$ from $\mathbb{Q}$ via the usual absolute value: one takes equivalence classes of Cauchy sequences and extends the absolute value by continuity. The valuation ring $\mathbb{Z}_p$ plays the role that $[-1, 1]$ plays in $\mathbb{R}$ — except that, thanks to the non-archimedean property, $\mathbb{Z}_p$ is actually a subring. To build intuition for the $p$-adic metric, it helps to see what convergence looks like in explicit examples. [example: $p$-Adic Convergence] In $\mathbb{Q}_5$ (the 5-adic numbers), the series $1 + 5 + 25 + 125 + \cdots = \sum_{n=0}^{\infty} 5^n$ converges because $|5^n|_5 = 5^{-n} \to 0$. Its sum is $1/(1 - 5) = -1/4$, demonstrating the remarkable fact that a series of positive integers can converge in $\mathbb{Q}_5$ to a rational number. More generally, for any prime $p$, the geometric series $\sum_{n=0}^{\infty} p^n$ converges in $\mathbb{Q}_p$ to $1/(1-p)$: \begin{align*} \frac{1}{1-p} = 1 + p + p^2 + p^3 + \cdots \end{align*} This is verified by noting that the partial sum $\sum_{n=0}^{N} p^n = (1 - p^{N+1})/(1-p)$, and $|p^{N+1}|_p = p^{-(N+1)} \to 0$. [/example] ### Structure of the $p$-Adic Integers We now investigate the internal structure of $\mathbb{Z}_p$. The first question is simply: what are its elements? The answer is that $\mathbb{Z}_p$ is precisely the closure of $\mathbb{Z}$ inside $\mathbb{Q}_p$ — so every $p$-adic integer is a limit of ordinary integers in the $p$-adic metric. [quotetheorem:2312] [citeproof:2312] This is a fundamental density result: every $p$-adic integer, however "exotic" it may seem as an infinite formal power series $\sum a_i p^i$, is approximated to arbitrary precision by ordinary integers. The field $\mathbb{Q}_p$ is in this sense a completion of $\mathbb{Q}$, completely analogous to how $\mathbb{R}$ is a completion of $\mathbb{Q}$ — just with a different notion of "closeness." [quotetheorem:2313] [citeproof:2313] The ideal structure of $\mathbb{Z}_p$ is dramatically simpler than that of $\mathbb{Z}$: whereas $\mathbb{Z}$ has one ideal $n\mathbb{Z}$ for every non-negative integer $n$, the ideals of $\mathbb{Z}_p$ form a single chain $\mathbb{Z}_p \supset p\mathbb{Z}_p \supset p^2\mathbb{Z}_p \supset \cdots$. The quotients $\mathbb{Z}_p/p^n\mathbb{Z}_p \cong \mathbb{Z}/p^n\mathbb{Z}$ show that modular arithmetic with $p$-power moduli is built directly into the structure of $\mathbb{Z}_p$. [quotetheorem:2314] The point of this result is the conceptual simplification: $\mathbb{Z}$ has infinitely many primes (2, 3, 5, 7, ...), but by passing to $\mathbb{Z}_p$ we "localise" and retain only the prime $p$, discarding all others. More precisely, any prime $q \neq p$ becomes a unit in $\mathbb{Z}_p$ (since $|q|_p = 1$, so $q \in \mathbb{Z}_p^\times$). The ring $\mathbb{Z}_p$ is a discrete valuation ring: a local PID with a unique maximal ideal $p\mathbb{Z}_p$. This is the algebraic structure that will generalise to the rings of integers of finite extensions of $\mathbb{Q}_p$. ### $\mathbb{Z}_p$ as a $p$-Adic Completion The two descriptions of $\mathbb{Z}_p$ — as a valuation ring inside $\mathbb{Q}_p$, and as the inverse limit $\varprojlim \mathbb{Z}/p^n\mathbb{Z}$ — are both fundamental, but they come from different directions. The first is analytic (completion of a metric space), the second is algebraic (compatible system of residues). That they agree is not obvious, and the following two theorems make this precise. We first show that the $p$-adic topology on $\mathbb{Z}$ (the $p\mathbb{Z}$-adic topology) coincides with the metric topology from $|\cdot|_p$. [quotetheorem:2377] [citeproof:2377] This theorem may seem pedantic — of course the two topologies agree, they are both "the $p$-adic topology" — but the point is that the $p\mathbb{Z}$-adic topology on $\mathbb{Z}$ is defined entirely in terms of ideals (no absolute value needed), while the metric topology is defined via $|\cdot|_p$. The fact that they coincide is a genuine content statement, not a tautology. [quotetheorem:2315] [citeproof:2315] This theorem closes the loop between the analytic and algebraic constructions of $\mathbb{Z}_p$. The surjectivity argument, in which we constructed the $p$-adic expansion digit by digit from a coherent sequence of residues, is genuinely constructive: it shows exactly how to represent an arbitrary element of $\mathbb{Z}_p$ as an infinite sum $\sum_{i=0}^{\infty} a_i p^i$. This representation is the key to explicit computation in $\mathbb{Z}_p$. ### $p$-Adic Expansions The surjectivity argument in the previous proof also establishes the fundamental structural theorem about $\mathbb{Z}_p$: [quotetheorem:2316] [citeproof:2316] The $p$-adic expansion theorem gives a concrete handle on elements of $\mathbb{Q}_p$: every $p$-adic number is a "Laurent series in $p$" with finitely many negative-power terms and digits in $\{0, 1, \ldots, p-1\}$. This representation makes arithmetic in $\mathbb{Q}_p$ computationally accessible, as the following example illustrates. [example: Negative Numbers in $\mathbb{Z}_p$] In $\mathbb{Z}_p$, the element $-1$ has the $p$-adic expansion $-1 = (p-1) + (p-1)p + (p-1)p^2 + \cdots$, i.e., all digits equal $p - 1$. This is analogous to the fact that $-1 = 0.\overline{9}$ in decimal. To verify: the partial sums $S_N = (p-1)\sum_{i=0}^{N} p^i = (p-1) \cdot \frac{p^{N+1}-1}{p-1} = p^{N+1} - 1 \equiv -1 \pmod{p^{N+1}}$, and $|S_N - (-1)|_p = |p^{N+1}|_p = p^{-(N+1)} \to 0$. The expansion $1/(1-p) = 1 + p + p^2 + \cdots$ follows from the same geometric series argument: the partial sums equal $(1 - p^{N+1})/(1-p)$, which differs from $1/(1-p)$ by $p^{N+1}/(1-p)$, and $|p^{N+1}/(1-p)|_p = p^{-(N+1)} \to 0$ since $|1-p|_p = 1$ (as $p \nmid (1-p)$ for $p > 2$; for $p = 2$, $|1-2|_2 = |-1|_2 = 1$ equally). [/example] The $p$-adic expansion should be thought of as the analogue of the decimal expansion of real numbers, but running to the left (with increasing powers of $p$) rather than to the right (with decreasing powers of $p$). Just as every real number has a decimal expansion, every $p$-adic integer has a $p$-adic expansion — but with the important difference that $p$-adic expansions are always eventually periodic exactly for elements of $\mathbb{Q} \cap \mathbb{Z}_p = \mathbb{Z}_{(p)}$, by a theorem analogous to the characterisation of rational numbers by periodic decimals. Having characterised all completions of Q through absolute values, we now build the algebraic and analytic machinery to study them rigorously. This chapter introduces absolute values on fields and develops the foundational properties that will underpin everything to follow. # 1. Basic Theory This chapter develops the general theory of valued fields, building on the concrete example of $\mathbb{Q}_p$ established in Chapter 1. The central questions are: what absolute values exist on a given field, and how do they behave under field extensions? Ostrowski's theorem gives a complete answer for $\mathbb{Q}$. The theory of completions makes precise what it means to "finish" a valued field, and the extension theorem tells us how to carry an absolute value across a finite extension. From these results we extract the definition of a local field and prove the fundamental classification theorem. ## Ostrowski's Theorem Before studying absolute values on extensions of $\mathbb{Q}$, we should understand all absolute values on $\mathbb{Q}$ itself. The answer is essentially forced: every non-trivial absolute value on $\mathbb{Q}$ is either the usual real absolute value (or a power thereof) or a $p$-adic absolute value for some prime $p$. [quotetheorem:2317] [citeproof:2317] Ostrowski's theorem is a uniqueness theorem at heart: it says that $\mathbb{Q}$ "sees" exactly one absolute value per prime $p$ (the $p$-adic one) plus one archimedean absolute value. The primes of $\mathbb{Q}$ are in bijection with the equivalence classes of non-trivial absolute values, if we include the "prime at infinity" corresponding to the archimedean absolute value. This is the starting point for the philosophy of the adeles: to study $\mathbb{Q}$ completely, one must study all its completions simultaneously. ## Completions of Valued Fields Given a valued field $(K, |\cdot|)$, we often cannot carry out limiting arguments directly in $K$ because $K$ may not be complete. The completion process resolves this by embedding $K$ into a larger field where all Cauchy sequences converge, while preserving the algebraic structure and extending the absolute value. [definition: Completion of a Valued Field] Let $(K, |\cdot|)$ be a valued field. A **completion** of $(K, |\cdot|)$ is a complete valued field $(\hat{K}, |\cdot|_{\hat{K}})$ together with an isometric field embedding $\iota: K \hookrightarrow \hat{K}$ such that $\iota(K)$ is dense in $\hat{K}$. [/definition] The condition that $\iota(K)$ is dense is essential: it says that $\hat{K}$ contains no "extra" elements beyond what is forced by the completion process. Without this, one could take $\hat{K}$ to be any large complete field and the definition would be vacuous. [quotetheorem:1002] [citeproof:1002] The Cauchy-sequence construction is entirely canonical, and it behaves well under field maps. [remark: Completions Are Functorial] The completion construction is functorial: a continuous field homomorphism $f: K \to L$ between valued fields extends uniquely to a continuous field homomorphism $\hat{f}: \hat{K} \to \hat{L}$. If $f$ is an isometry, then $\hat{f}$ is an isometry. In particular, automorphisms of $K$ that preserve the absolute value extend to automorphisms of $\hat{K}$. [/remark] [example: Completions of $\mathbb{Q}$] By Ostrowski's theorem, the completions of $\mathbb{Q}$ are: - $\mathbb{R}$, the completion with respect to $|\cdot|_\infty$; - $\mathbb{Q}_p$, the completion with respect to $|\cdot|_p$, for each prime $p$. These are the only completions of $\mathbb{Q}$ up to isometric isomorphism. This is why local fields are ubiquitous in number theory: any analytic study of $\mathbb{Q}$ at a prime $p$ naturally lands in $\mathbb{Q}_p$. [/example] ## Valuations and the Valuation Ring Structure Working with absolute values, divisibility relationships become obscured by multiplicative notation -- the fact that $p^3$ divides $x$ is encoded as $|x|_p \leq p^{-3}$, which is harder to read than $v_p(x) \geq 3$. How can we reformulate the theory additively? The answer is to pass from the absolute value $|\cdot|$ to the **valuation** $v = -\log_c|\cdot|$, which converts the multiplicative structure into an additive one. The two perspectives carry identical information, but the valuation turns divisibility statements into transparent inequalities. [definition: Valuation] Let $K$ be a field. A **valuation** on $K$ is a function $v: K \to \mathbb{R} \cup \{\infty\}$ satisfying: 1. $v(x) = \infty$ if and only if $x = 0$; 2. $v(xy) = v(x) + v(y)$ for all $x, y \in K$; 3. $v(x + y) \geq \min\{v(x), v(y)\}$ for all $x, y \in K$. We use the conventions $r + \infty = \infty$ and $r \leq \infty$ for all $r \in \mathbb{R}$. [/definition] The passage between valuations and absolute values is immediate: given a valuation $v$, the function $|x| = c^{-v(x)}$ for any $c > 1$ is a non-archimedean absolute value. Conversely, $v(x) = -\log_c|x|$ recovers the valuation from the absolute value. One reason to prefer valuations is that in certain situations there is a canonical normalisation of $v$ — for example, $v_p$ on $\mathbb{Q}_p$ with $v_p(p) = 1$ — whereas the absolute value requires an arbitrary choice of the base $c$. [example: The $p$-Adic Valuation] On $\mathbb{Q}_p$, the **$p$-adic valuation** is $v_p(x) = -\log_p|x|_p$. For $x \in \mathbb{Z}_p$, we have $v_p(x) = n$ iff $p^n \| x$ (meaning $p^n$ divides $x$ but $p^{n+1}$ does not). The normalisation $v_p(p) = 1$ is canonical. The absolute value is recovered as $|x|_p = p^{-v_p(x)}$. [/example] The same valuation formalism applies equally well over function fields, where the role of $p$ is played by the variable $T$. [example: Formal Laurent Series] Let $k$ be a field and consider the field of **formal Laurent series** \begin{align*} k((T)) = \left\{ \sum_{i=n}^{\infty} a_i T^i : a_i \in k,\, n \in \mathbb{Z} \right\}. \end{align*} Define $v\!\left(\sum a_i T^i\right) = \min\{i : a_i \neq 0\}$. This is a valuation on $k((T))$: $v$ is infinite exactly on zero, $v(fg) = v(f) + v(g)$ follows from the leading-term behaviour of products, and $v(f + g) \geq \min\{v(f), v(g)\}$ follows from the definition of minimum. The valuation ring is $k[[T]]$, the formal power series, and the residue field is $k$ itself. [/example] The valuation ring, maximal ideal, and residue field associated to a non-archimedean valued field are the key invariants. Recall from Chapter 1 that for a non-archimedean valued field $K$, the valuation ring is $\mathcal{O}_K = \{x \in K : |x| \leq 1\}$. In the valuation language, $\mathcal{O}_K = \{x : v(x) \geq 0\}$. The **maximal ideal** is $\mathfrak{m}_K = \{x \in K : |x| < 1\} = \{x : v(x) > 0\}$, and it is indeed the unique maximal ideal of $\mathcal{O}_K$: if $x \notin \mathfrak{m}_K$, then $|x| = 1$, so $x^{-1} \in \mathcal{O}_K$ and $x$ is a unit. [definition: Residue Field] The **residue field** of a non-archimedean valued field $K$ is the quotient \begin{align*} k_K = \mathcal{O}_K / \mathfrak{m}_K. \end{align*} [/definition] [example: Residue Fields of $\mathbb{Q}_p$ and $k((T))$] For $K = \mathbb{Q}_p$, the valuation ring is $\mathbb{Z}_p$, the maximal ideal is $p\mathbb{Z}_p$, and the residue field is $\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{Z}/p\mathbb{Z} = \mathbb{F}_p$. For $K = k((T))$, the valuation ring is $k[[T]]$, the maximal ideal is $T \cdot k[[T]]$, and the residue field is $k[[T]]/(T) \cong k$. These two examples illustrate the two types of local fields we will classify below. [/example] ## Hensel's Lemma Given a polynomial $f \in K[x]$ over a non-archimedean valued field $K$, how can we find roots of $f$ in $K$ or its ring of integers? Working directly in $K$ is hard. But if $K$ has a residue field $k_K$, we can reduce $f$ to $\bar{f} \in k_K[x]$ and find roots there — a much simpler algebraic problem. The question is whether a root of $\bar{f}$ can be promoted to a genuine root of $f$. Over $\mathbb{Q}$, this fails: $x^2 - 2$ has the root $3$ modulo $7$ (since $3^2 = 9 \equiv 2 \pmod{7}$), yet $\sqrt{2} \notin \mathbb{Q}$. The key property that $\mathbb{Q}$ lacks and that $\mathbb{Q}_7$ possesses is completeness. Hensel's lemma says that over a complete non-archimedean field, a simple root of $\bar{f}$ always lifts to a root of $f$. One of the defining features of complete non-archimedean fields is Hensel's lemma, which allows us to lift factorisations from the residue field back to the original field. This is the non-archimedean analogue of Newton's method: if a polynomial has a "good" factorisation modulo the maximal ideal, that factorisation lifts uniquely to the whole ring. For a non-archimedean valued field $K$ and $f \in K[x]$, we call $f$ **primitive** if $\max_i |a_i| = 1$, where $a_0, \ldots, a_n$ are the coefficients. Equivalently, $f \in \mathcal{O}_K[x]$ and at least one coefficient is a unit. A primitive polynomial has a well-defined reduction $\bar{f} = f \bmod \mathfrak{m}_K \in k_K[x]$, which is non-zero. [quotetheorem:2318] [citeproof:2318] Hensel's lemma is one of the most powerful results in the course. Its impact is felt immediately: it says that roots of polynomials over the residue field — a much simpler algebraic object — lift to roots over $K$. This is entirely unlike the archimedean case, where there is no systematic way to lift from $\mathbb{R}/\mathbb{Z}$ to $\mathbb{R}$. [quotetheorem:2319] [citeproof:2319] [example: Roots of Unity in $\mathbb{Z}_p$] The polynomial $x^{p-1} - 1 \in \mathbb{Z}_p[x]$ reduces to $x^{p-1} - 1 \in \mathbb{F}_p[x]$. By Fermat's little theorem, every non-zero element of $\mathbb{F}_p$ is a root of $x^{p-1} - 1$, and since $\mathbb{F}_p^\times$ is cyclic of order $p-1$, the polynomial has $p-1$ distinct (simple) roots in $\mathbb{F}_p$. Applying the corollary to each root, $\mathbb{Z}_p$ contains all $(p-1)$-th roots of unity, i.e., all primitive $d$-th roots of unity for $d \mid p-1$. In particular, $\mathbb{Z}_p$ contains a full set of representatives for $\mathbb{F}_p^\times$, which are called the **Teichmüller representatives**. [/example] [example: Square Roots in $\mathbb{Q}_7$] The element $2 \in \mathbb{F}_7$ is a quadratic residue (since $3^2 = 9 \equiv 2 \pmod{7}$), so $x^2 - 2$ has a simple root in $\mathbb{F}_7$. By the corollary, $\sqrt{2} \in \mathbb{Q}_7$. [/example] The irreducibility criterion is a useful consequence, and it is here that the completeness hypothesis becomes visible: without it, the Hensel lifting step used in the proof would not work. [quotetheorem:2320] [citeproof:2320] This theorem quantifies the "spread" of coefficient sizes in an irreducible polynomial, and it is geometrically transparent via Newton polygons: the condition $|a_\ell| \leq \max(|a_0|, |a_n|)$ for all $\ell$ is exactly the condition that the Newton polygon has no interior points strictly below the segment connecting its endpoints. We will return to this when we study Newton polygons in full generality below. ## Extending Absolute Values to Finite Extensions The most important application of Hensel's lemma — and the key step toward the classification of local fields — is the extension theorem for absolute values. Given a complete valued field $K$ and a finite extension $L/K$, we wish to extend the absolute value from $K$ to $L$. Unlike the situation over $\mathbb{Q}$ (where an extension such as $\mathbb{Q}(\sqrt{2})$ admits multiple distinct absolute values), over a complete field the extension is unique. Before we can prove the main theorem, we need to understand norms on vector spaces over complete valued fields. [definition: Non-Archimedean Norm on a Vector Space] Let $K$ be a non-archimedean valued field and $V$ a $K$-vector space. A **norm** on $V$ is a function $\|\cdot\|: V \to \mathbb{R}_{\geq 0}$ satisfying: 1. $\|x\| = 0$ iff $x = 0$; 2. $\|\lambda x\| = |\lambda| \cdot \|x\|$ for all $\lambda \in K$, $x \in V$; 3. $\|x + y\| \leq \max(\|x\|, \|y\|)$ (the non-archimedean strong triangle inequality). Two norms $\|\cdot\|$ and $\|\cdot\|'$ on $V$ are **equivalent** if there exist $C, D > 0$ with $C\|x\| \leq \|x\|' \leq D\|x\|$ for all $x \in V$. [/definition] The max norm on a finite-dimensional $K$-vector space gives an explicit and convenient norm. Fix a basis $x_1, \ldots, x_n$ of $V$; for $x = \sum a_i x_i$ define $\|x\|_{\max} = \max_i |a_i|$. This is a norm because the strong triangle inequality holds coordinate-wise. [quotetheorem:2321] [citeproof:2321] This result is striking: the topology on a finite-dimensional space over a complete valued field is unique — no matter which norm you put on it, you get the same open sets and the same notion of convergence. Completeness is essential: over $\mathbb{Q}_p$ with an incomplete base field, one can construct inequivalent norms on infinite-dimensional spaces, and the theorem fails for finite-dimensional spaces over non-complete fields in characteristic $p$. A key consequence is that $L$ is automatically complete under any absolute value extending that of $K$ — this is what we use in the uniqueness argument for the extension theorem. We also need two lemmas for the main theorem. The first says that valuation rings are integrally closed. [quotetheorem:2322] [citeproof:2322] Note that this would fail for archimedean fields: in $\mathbb{R}$, the "valuation ring" $\{x : |x| \leq 1\} = [-1, 1]$ is not even a ring, let alone integrally closed. The non-archimedean strong triangle inequality is exactly what makes the valuation ring closed under addition and hence an actual ring. This lemma is central to the extension theorem: we use it to identify the valuation ring $\mathcal{O}_L$ of the extended absolute value as the integral closure of $\mathcal{O}_K$ in $L$. [quotetheorem:2323] [citeproof:2323] The value of this criterion is that it collapses a universal quantification — checking the strong triangle inequality for all pairs $x, y$ — to a single-point condition at $0$ and $1$. This makes it much easier to verify non-archimedean behaviour in new settings, and it is precisely what we need in the extension theorem to check that $|\cdot|_L = |N_{L/K}(\cdot)|^{1/n}$ satisfies the strong triangle inequality without doing a direct computation with Galois conjugates. We now have all the tools for the main theorem. [quotetheorem:2324] [citeproof:2324] [remark: The Integral Closure Corollary] The proof shows a bonus: for a complete valued field $K$ with finite extension $L/K$ equipped with the extended absolute value $|\cdot|_L$, the valuation ring $\mathcal{O}_L$ is exactly the integral closure of $\mathcal{O}_K$ in $L$. This is a fundamental fact that will be used repeatedly in the ramification theory of Chapter 5. [/remark] The uniqueness of the extension also has a striking consequence for the Galois group: all automorphisms of $L/K$ must preserve the absolute value. [quotetheorem:2325] [citeproof:2325] This theorem reflects a deep principle: the metric structure on $L$ is an intrinsic property of the extension $L/K$, not dependent on any particular embedding of $L$. ## Local Fields: Definition and Classification Which complete non-archimedean valued fields are arithmetically interesting? An arbitrary complete field can be enormous — for instance, $\mathbb{C}_p$, the completion of the algebraic closure $\overline{\mathbb{Q}_p}$, is complete and algebraically closed, but its residue field is the algebraic closure $\overline{\mathbb{F}_p}$, which is infinite and unwieldy. This makes $\mathbb{C}_p$ hard to classify and computationally difficult. If we demand that the residue field be finite, the theory becomes tractable and the classification is complete. Such fields are called local fields, and they are precisely the $p$-adic fields $\mathbb{Q}_p$ and their finite extensions (in characteristic 0) and the formal Laurent series fields over finite fields (in characteristic $p$). [definition: Local Field] A **local field** is a non-archimedean valued field $(K, |\cdot|)$ that is: 1. complete with respect to $|\cdot|$, and 2. has finite residue field $k_K = \mathcal{O}_K/\mathfrak{m}_K$. [/definition] The name "local field" comes from a geometric and algebraic philosophy worth pausing to appreciate. [remark: Why "Local"?] The term "local" refers to the philosophy of localisation in algebraic number theory. If $F$ is a number field (a finite extension of $\mathbb{Q}$) and $\mathfrak{p}$ is a prime ideal of its ring of integers $\mathcal{O}_F$, the completion of $F$ at $\mathfrak{p}$ is a local field. Studying $F$ "locally at $\mathfrak{p}$" means studying this completion. The global field $F$ is recovered (in a suitable sense) from the product of all its local completions — this is the adelic perspective. [/remark] The next step is to understand the internal structure of a local field $K$. Since $k_K$ is finite of characteristic $p$, we have $k_K = \mathbb{F}_{p^f}$ for some prime $p$ and $f \geq 1$. The prime $p$ is the **characteristic of the residue field**. There are two fundamentally different cases depending on the characteristic of $K$ itself. **Characteristic 0 case.** If $\operatorname{char}(K) = 0$, then $K$ contains $\mathbb{Q}$, and the restriction of $|\cdot|$ to $\mathbb{Q}$ is non-trivial. By Ostrowski's theorem, $|\cdot|_{\mathbb{Q}} = |\cdot|_p^s$ for some prime $p$ and $s > 0$. Thus $K$ is an extension of $\mathbb{Q}_p$, and since $K$ is complete, it contains the completion of $\mathbb{Q}$ in $|\cdot|_p$, which is $\mathbb{Q}_p$. So $K$ is a finite extension of $\mathbb{Q}_p$. **Characteristic $p$ case.** If $\operatorname{char}(K) = p$, then $K$ contains $\mathbb{F}_p$. Since $k_K = \mathbb{F}_{p^f}$ and the residue field is the quotient $\mathcal{O}_K/\mathfrak{m}_K$, the ring $\mathcal{O}_K$ contains a lift of $k_K$. Let $\pi$ be a uniformiser (a generator of $\mathfrak{m}_K$). Every element of $\mathcal{O}_K$ has a unique $\pi$-adic expansion with digits in a fixed set of representatives for $k_K$, and $K = k_K((\pi))$, the formal Laurent series field. More precisely, $K \cong \mathbb{F}_{p^f}((\pi))$. [quotetheorem:2326] The proof of this theorem in full generality requires the structure theory of complete discrete valuation rings, which we develop in the next chapter. The argument above shows the dichotomy: characteristic 0 forces $K/\mathbb{Q}_p$ finite, and characteristic $p$ forces $K \cong \mathbb{F}_q((T))$. This classification is exhaustive: there are no "exotic" local fields beyond these two families. Every local field is built from a prime $p$ and a finite residue field $\mathbb{F}_q$, and the classification pins down the structure completely. The finiteness of the residue field is essential -- dropping it admits fields like $\overline{\mathbb{F}_p}((T))$, whose residue field $\overline{\mathbb{F}_p}$ is algebraically closed but infinite, and such fields have much wilder behaviour (for instance, the Galois theory of their extensions is far less constrained). The archimedean local fields $\mathbb{R}$ and $\mathbb{C}$ are excluded from this classification by convention: they are the completions of $\mathbb{Q}$ at the archimedean place, and their theory is classical analysis rather than $p$-adic analysis. Including them would make "local field" synonymous with "locally compact non-discrete topological field," which is the broader classification due to Pontryagin. Looking ahead, the significance of local fields extends far beyond their internal structure: local class field theory (Chapter 7) gives a complete description of all abelian extensions of a local field $K$ in terms of the multiplicative group $K^\times$, and this local theory is a crucial ingredient in the proof of global class field theory for number fields. [example: The Two Families of Local Fields] The prime $p = 5$ gives: in characteristic 0, the fields $\mathbb{Q}_5$, $\mathbb{Q}_5(\sqrt{5})$, $\mathbb{Q}_5(\zeta_5)$ (where $\zeta_5$ is a primitive 5th root of unity), and all other finite extensions; in characteristic 5, the fields $\mathbb{F}_5((T))$, $\mathbb{F}_{25}((T))$, $\mathbb{F}_{125}((T))$, and so on. Characteristic 0 and characteristic $p$ local fields are fundamentally different: in characteristic 0, $K$ contains a copy of $\mathbb{Z}_p$ whose ring of integers has characteristic 0 even though the residue field has characteristic $p$. In characteristic $p$, the entire ring $\mathcal{O}_K = \mathbb{F}_q[[T]]$ has characteristic $p$. [/example] ## Newton Polygons Given a polynomial $f \in K[x]$ over a complete valued field, how can we determine the valuations of its roots without factoring $f$ — which may be difficult or impossible to do explicitly? The valuations of the roots are determined entirely by the valuations of the coefficients, and Newton polygons make this dependency geometric. The construction organises the points $(i, v(a_i))$ in the plane into a piecewise-linear boundary — the lower convex hull — whose slopes record the valuations of the roots. This is a strikingly non-algebraic tool: it translates a question about arithmetic (where are the roots?) into a question about geometry (what does the lower boundary of a point set look like?). We work with the valuation $v$ rather than the absolute value. Given $f(x) = a_0 + a_1 x + \cdots + a_n x^n \in K[x]$, we want to organise the valuations $v(a_i)$ to read off the valuations of the roots of $f$. [definition: Lower Convex Set and Lower Convex Hull] A set $S \subseteq \mathbb{R}^2$ is **lower convex** if: 1. whenever $(x, y) \in S$, then $(x, z) \in S$ for all $z \geq y$; and 2. $S$ is convex. Given any set of points $T \subseteq \mathbb{R}^2$, the minimal lower convex set containing $T$ is the **lower convex hull** of $T$. [/definition] With this geometric setup in place, the Newton polygon of a polynomial is simply the lower convex hull of its coefficient data. [definition: Newton Polygon] Let $f(x) = a_0 + a_1 x + \cdots + a_n x^n \in K[x]$ with $(K, v)$ a valued field. The **Newton polygon** of $f$ is the lower convex hull of the set of points $\{(i, v(a_i)) : 0 \leq i \leq n,\, a_i \neq 0\}$. [/definition] In practice, the Newton polygon is described by its **break points** — the points $(i, v(a_i))$ lying on the lower boundary — and the **line segments** connecting adjacent break points. Each line segment has a well-defined **slope** and **length** (horizontal run).  [example: Newton Polygon of a Polynomial over $\mathbb{Q}_p$] Consider $(\mathbb{Q}_p, v_p)$ with the polynomial $f(x) = p^3 + px + p^3 x^2 - p^2 x^3 + p^2 x^4 + x^5$. The valuations of the coefficients at positions $0, 1, 2, 3, 4, 5$ are $3, 1, 3, 2, 2, 0$. Plotting $(i, v(a_i))$ at $(0,3), (1,1), (2,3), (3,2), (4,2), (5,0)$, the lower convex hull has break points at $(0,3), (1,1), (5,0)$. The two line segments have slopes $(1-3)/(1-0) = -2$ and $(0-1)/(5-1) = -1/4$, with lengths $1$ and $4$ respectively. [/example] The Newton polygon is not merely decorative — it encodes the valuations of all the roots. [quotetheorem:2378] [citeproof:2378] The theorem counts roots with multiplicity, so a segment of horizontal length $s - r$ accounts for $s - r$ roots (in the splitting field). The remarkable fact is that this count depends only on the valuations $v(a_i)$, not on the actual coefficients — this is a manifestation of the non-archimedean ultrametric property, where exact cancellations are impossible between terms of different valuation. The slopes of the Newton polygon are precisely the negatives of the root valuations, connecting Newton polygons to the tropical geometry perspective where one works in $(\mathbb{R}, \min, +)$ rather than $(\mathbb{R}, +, \times)$. [quotetheorem:2327] [citeproof:2327] [remark: Eisenstein's Criterion] The converse direction provides the link to Eisenstein's criterion. A polynomial $f(x) = a_n x^n + \cdots + a_0$ is Eisenstein at $p$ if $v(a_n) = 0$, $v(a_i) \geq 1$ for $i < n$, and $v(a_0) = 1$. Its Newton polygon has break points at $(0, 1)$ and $(n, 0)$, giving a single segment of slope $-1/n$. The theorem says all roots have the same valuation $1/n$, which forces $f$ to be irreducible over $K$ (since any factorisation would produce a factor of degree $< n$ whose Newton polygon would have a single segment of slope $-1/n$, but such a slope in an integer factorisation is impossible unless the factor is linear with an irrational-valued root — more precisely, any proper factor would have all its roots of valuation $1/n$, giving a coefficient of absolute value $p^{-k/n}$ for the constant term of a degree-$k$ factor, and this is not in $K$ unless $n \mid k$). [/remark] ## Structure of Completions of Number Fields What local fields arise from number fields? If $F/\mathbb{Q}$ is a finite extension and $\mathfrak{p}$ is a prime of $\mathcal{O}_F$, the completion $F_\mathfrak{p}$ is a local field by our classification. But which one? The answer is controlled by two invariants: the ramification index $e_\mathfrak{p}$, which measures how many times $p$ divides in $\mathfrak{p}$, and the inertia degree $f_\mathfrak{p}$, which measures the size of the residue field. Together they determine the degree of $F_\mathfrak{p}$ over $\mathbb{Q}_p$, and the fundamental identity $\sum_{\mathfrak{p} \mid p} e_\mathfrak{p} f_\mathfrak{p} = [F:\mathbb{Q}]$ ties the local picture back to the global degree. We close this chapter by placing local fields in their arithmetic context. The completions of a number field at its finite places are the local fields attached to the field, and their structure is controlled by the prime $\mathfrak{p}$ below. Let $F$ be a number field and $\mathfrak{p}$ a prime ideal of $\mathcal{O}_F$ (lying over the rational prime $p$). The $\mathfrak{p}$-adic absolute value on $F$ is defined by $|x|_\mathfrak{p} = N(\mathfrak{p})^{-v_\mathfrak{p}(x)}$, where $N(\mathfrak{p}) = |\mathcal{O}_F/\mathfrak{p}|$ is the absolute norm and $v_\mathfrak{p}$ is the $\mathfrak{p}$-adic valuation. The completion $F_\mathfrak{p}$ with respect to this absolute value is a local field. [quotetheorem:2328] The local degree $e_\mathfrak{p} f_\mathfrak{p}$ is the contribution of $\mathfrak{p}$ to the factorisation $(p) = \prod_{\mathfrak{p} \mid p} \mathfrak{p}^{e_\mathfrak{p}}$, and the familiar fundamental identity $\sum_{\mathfrak{p} \mid p} e_\mathfrak{p} f_\mathfrak{p} = [F:\mathbb{Q}]$ says that the sum of local degrees over all primes above $p$ equals the global degree. This theorem applies only to the non-archimedean (finite) places of $F$. At the archimedean places, the completions are $\mathbb{R}$ or $\mathbb{C}$: a real embedding $F \hookrightarrow \mathbb{R}$ gives a completion isomorphic to $\mathbb{R}$, while a pair of conjugate complex embeddings gives a completion isomorphic to $\mathbb{C}$. These archimedean completions are not local fields in our sense, but they are essential for the full local-global picture (the product formula, the adele ring, and the Hasse-Minkowski theorem all require both finite and infinite places). One should also note that the theorem requires $F/\mathbb{Q}$ to be a *finite* extension: for an infinite algebraic extension such as $\mathbb{Q}(\zeta_{p^\infty}) = \bigcup_n \mathbb{Q}(\zeta_{p^n})$, the completion at a prime above $p$ is no longer a finite extension of $\mathbb{Q}_p$, and the resulting field is not locally compact. [example: Completions of $\mathbb{Q}(\sqrt{-5})$ at Primes above 2] Let $F = \mathbb{Q}(\sqrt{-5})$. The prime $2$ factors in $\mathcal{O}_F = \mathbb{Z}[\sqrt{-5}]$ as $(2) = (2, 1 + \sqrt{-5})^2$, a single prime squared. So there is one prime $\mathfrak{p} = (2, 1 + \sqrt{-5})$ above $2$, with $e_\mathfrak{p} = 2$ and $f_\mathfrak{p} = 1$. The completion $F_\mathfrak{p}$ is a ramified extension of $\mathbb{Q}_2$ of degree $2 \cdot 1 = 2$, with residue field $\mathbb{F}_2$. In contrast, the prime $3$ in $F$ factors as $(3) = (3, 1 + \sqrt{-5})(3, 1 - \sqrt{-5})$, two distinct primes each with $e = 1$ and $f = 1$. The completions $F_\mathfrak{p}$ and $F_{\mathfrak{p}'}$ are each isomorphic to $\mathbb{Q}_3$. [/example] The study of $F$ "at $\mathfrak{p}$" means the study of the local field $F_\mathfrak{p}$. The ramification index $e$ and inertia degree $f$ are the two fundamental invariants of this local extension, and their product equals the local degree. Understanding $e$ and $f$ is the subject of the ramification theory developed in Chapters 5 and 6. The p-adic rationals Q_p exemplify a complete valued field with periodic expansions, yet they are just one instance of a much broader class. This chapter generalises from Q_p to develop the theory of valued fields, establishing the framework in which all our subsequent work will be framed. # 2. Valued Fields Chapter 2 established the general framework of complete non-archimedean valued fields, proved Hensel's lemma, and showed how absolute values extend uniquely to finite extensions. Armed with these tools, we now specialise to the class of fields whose valuations take values in $\mathbb{Z}$ — the **discretely valued fields**. This extra finiteness condition on the value group has dramatic structural consequences: the valuation ring becomes a principal ideal domain, every element has a $\pi$-adic expansion, and the field is essentially determined by its residue field together with a uniformiser. The central results of this chapter are the structure theorem classifying complete DVFs of equal characteristic via Teichmüller lifts, and its mixed-characteristic counterpart via Witt vectors — tools that explain precisely why $\mathbb{Q}_p$ and $\mathbb{F}_p((T))$ are the two archetypal local fields. ## Discrete Valuations and DVRs In Chapter 2, we worked with general valued fields whose value group $v(K^\times)$ could be any subgroup of $\mathbb{R}$. Most of the fields we actually care about — $\mathbb{Q}_p$, $\mathbb{F}_q((T))$, and their finite extensions — have a much more constrained value group: it is discrete in $\mathbb{R}$. A discrete subgroup of $\mathbb{R}$ is necessarily cyclic, so $v(K^\times)$ is either the zero subgroup or isomorphic to $\mathbb{Z}$. [definition: Discretely Valued Field] Let $K$ be a valued field with valuation $v: K \to \mathbb{R} \cup \{\infty\}$. We say $K$ is a **discretely valued field** (DVF) if $v(K^\times)$ is a discrete subgroup of $\mathbb{R}$, i.e., $v(K^\times)$ is infinite cyclic. [/definition] Note that we do not require $v(K^\times) = \mathbb{Z}$ — we allow any infinite cyclic subgroup. This flexibility matters for field extensions: the value group of an extension of a DVF is not always $\mathbb{Z}$, though it is always a discrete subgroup of $\mathbb{R}$. For computation, however, there is always a canonical choice of valuation that does land in $\mathbb{Z}$. [definition: Normalized Valuation] Let $K$ be a DVF. The **normalized valuation** $v_K$ on $K$ is the unique valuation on $K$, in the given equivalence class, whose image is exactly $\mathbb{Z}$. [/definition] The normalized valuation does not determine a canonical absolute value, since one must still choose a base $c > 1$ to set $|x| = c^{-v_K(x)}$. The real-valued absolute value carries redundant data (the choice of $c$), while the $\mathbb{Z}$-valued normalized valuation strips away this redundancy. [definition: Uniformizer] Let $K$ be a DVF with normalized valuation $v_K$. An element $\pi \in K$ is a **uniformizer** if $v_K(\pi) = 1$, i.e., if $\pi$ has the smallest positive valuation. [/definition] The uniformizer generates the maximal ideal $\mathfrak{m}_K = \pi \mathcal{O}_K$. Two uniformizers differ by a unit in $\mathcal{O}_K$. [example: Uniformizers in Ramified Extensions] The usual valuation on $\mathbb{Q}_p$ is normalized, with $v_p(p) = 1$, so $p$ is the uniformizer for $\mathbb{Q}_p$. Now adjoin a root of $f(X) = X^2 - p$ to form $K = \mathbb{Q}_p(\sqrt{p})$. The extension $K/\mathbb{Q}_p$ has degree $2$ and the normalized valuation on $K$ satisfies $v_K(\sqrt{p}) = 1$ (since $v_K(p) = v_K((\sqrt{p})^2) = 2$). So $\sqrt{p}$, not $p$, is the uniformizer for $K$; the prime $p = (\sqrt{p})^2$ satisfies $v_K(p) = 2$, meaning $p$ has ramified in the extension. The valuation ring is $\mathcal{O}_K = \mathbb{Z}_p[\sqrt{p}]$ and the unique maximal ideal is $\mathfrak{m}_K = (\sqrt{p})$, while $(p) = \mathfrak{m}_K^2$. This illustrates how extensions can change the uniformizer: the prime that generated $\mathfrak{m}_{\mathbb{Q}_p}$ now generates $\mathfrak{m}_K^2$, and the new uniformizer has valuation $1$ in the renormalized scale. [/example] The algebraic structure of the valuation ring of a DVF is particularly clean. [definition: Discrete Valuation Ring] A ring $R$ is a **discrete valuation ring** (DVR) if it is a principal ideal domain (PID) with a unique prime element up to units. [/definition] This algebraic characterisation is equivalent to being the valuation ring of a DVF. [quotetheorem:2379] [citeproof:2379] The proposition implies that the theory of DVRs and the theory of DVFs are the same theory viewed from two different angles. The algebraic perspective (DVR) emphasises the ideal-theoretic structure, while the valuation-theoretic perspective (DVF) provides the analytic machinery. If $R$ were merely a local ring or a valuation ring without the PID condition, the equivalence would break: general valuation rings need not have principal maximal ideals, and the uniformizer — the generator giving the $\mathbb{Z}$-graded filtration — would not exist. The DVR condition is what makes every ideal a power of $\mathfrak{m}$, collapsing the entire ideal theory to a single element $\pi$. The key structural properties of a DVF, assembled into one statement, are as follows. [quotetheorem:2329] [citeproof:2329] Property (5) is the most striking: a complete DVF looks exactly like a formal Laurent series field, with $\pi$ playing the role of the variable and $S$ playing the role of the coefficient field. This is not a coincidence — it is the content of the structure theorem we will prove below. Two features of the completeness assumption are worth isolating. For an incomplete DVF, the $\pi$-adic expansions of elements in $K$ still exist, but they need not converge back into $K$: incompleteness means that Cauchy sequences in $K$ can escape to the completion $\hat{K}$, so $\pi$-adic series are naturally elements of $\hat{K}$, not $K$ itself. The discreteness of $v_K$ is what singles out a single prime element $\pi$ and makes the filtration $\mathfrak{m}^n = \pi^n \mathcal{O}_K$ a simple arithmetic progression; for a non-discrete valuation the filtration ideals $\{x : v(x) > r\}$ do not have a smallest element and the expansion in part (5) has no natural digit basis. The compactness of $\mathcal{O}_K$ is a clean necessary and sufficient condition for $K$ to be a local field. The theorem shows that compactness is the correct algebraic-analytic boundary between local fields and more general complete DVFs. To see why infinite residue fields fail: $\mathbb{C}((T))$ is a complete DVF (equal characteristic $0$, uniformizer $T$, residue field $\mathbb{C}$) but $\mathcal{O} = \mathbb{C}[[T]]$ is not compact — the open cover $\{B(f, |T|) : f \in \mathcal{O}\}$ has no finite subcover, as the residue classes $\mathcal{O}/T\mathcal{O} \cong \mathbb{C}$ are uncountably many. Compactness forces both completeness and finiteness of the residue field simultaneously: it is not possible to have one without the other. [quotetheorem:2330] [citeproof:2330] The compactness criterion shows that the local field condition is topological in nature: it is the exactly the statement that $\mathcal{O}_K$ is a compact topological ring. When the residue field is infinite, compactness fails in a very explicit way — for $\mathbb{C}((T))$, the valuation ring $\mathbb{C}[[T]]$ is complete and the valuation is discrete, yet the uncountably many residue classes $\mathcal{O}/T\mathcal{O} \cong \mathbb{C}$ prevent any finite subcover from existing, so completeness alone does not force compactness. Conversely, one might ask whether a compact $\mathcal{O}_K$ could fail to come from a DVF: the answer is no, because compactness of a valuation ring in a non-archimedean valued field already forces discreteness of the value group (the maximal ideal must be finitely generated, hence principal). This criterion will play a central role in the ramification theory of Chapter 4, where compactness of $\mathcal{O}_K$ ensures that residue field extensions are finite and the fundamental identity $efn = [L:K]$ holds. ## Equal and Mixed Characteristic The structure theorems we aim to prove split into two fundamentally different cases. Why does this split occur, and what goes wrong if we try a uniform approach? Here is the difficulty. In both $\mathbb{Q}_p$ and $\mathbb{F}_p((T))$, the residue field is $\mathbb{F}_p$ and the valuation ring has a $\pi$-adic expansion. If the two cases were uniform, we should be able to use the same construction to lift $\mathbb{F}_p$ to $\mathcal{O}_K$ in both cases. But in $\mathbb{F}_p((T))$ we have $\operatorname{char}(\mathcal{O}_K) = p$, so the Teichmüller lift is a ring homomorphism (all cross terms in the binomial expansion of $(a+b)^p$ vanish). In $\mathbb{Z}_p$ we have $\operatorname{char}(\mathcal{O}_K) = 0$, so $p \neq 0$ in $\mathcal{O}_K$ and the Teichmüller lift is only multiplicative — additivity fails. For a concrete failure: in $\mathbb{Z}_p$ one checks $[1] + [1] = 1 + 1 = 2$, yet $[2] = 2$ only when $p > 2$; and more strikingly, for $p = 2$ one has $[1] + [1] = 2 \neq 0 = [0]$ even though $1 + 1 = 0$ in $\mathbb{F}_2$. The Teichmüller lift fails to be a ring homomorphism in mixed characteristic: there is no way to make it additive without collapsing the characteristic. This is why the mixed characteristic case requires a genuinely different construction — the Witt vectors — that encodes how the carries in addition interact with the prime $p$. The two most important examples of local fields, $\mathbb{Q}_p$ and $\mathbb{F}_p((T))$, both have residue field $\mathbb{F}_p$, but they differ fundamentally in one respect: $\mathbb{F}_p((T))$ has characteristic $p$ as a field, while $\mathbb{Q}_p$ has characteristic $0$. This dichotomy is a fundamental invariant of DVFs. [definition: Equal and Mixed Characteristic] Let $K$ be a valued field with residue field $k_K$. We say $K$ has **equal characteristic** if $\operatorname{char}(K) = \operatorname{char}(k_K)$, and **mixed characteristic** if $\operatorname{char}(K) \neq \operatorname{char}(k_K)$. [/definition] In the mixed characteristic case, necessarily $\operatorname{char}(K) = 0$ and $\operatorname{char}(k_K) = p > 0$, since a field of characteristic $p > 0$ always passes characteristic $p$ to any quotient. [example: Characteristic Does Not Pass Through Extensions Uniformly] Let $K = \mathbb{Q}_p(\sqrt{p})$. The element $\sqrt{p}$ is a uniformizer for $K$ (since $v_K(\sqrt{p}) = 1$ in the normalized valuation of $K$, while $v_K(p) = 2$). The residue field of $K$ is still $\mathbb{F}_p$ (any root of a unit over $\mathbb{F}_p$ is a unit, and $\sqrt{p} \equiv 0 \pmod{\mathfrak{m}_K}$). So $K$ has mixed characteristic: $\operatorname{char}(K) = 0$ and $\operatorname{char}(k_K) = p$. Now consider the equal characteristic analogue $k((T^{1/2}))$ over a field $k$ of characteristic $p$: the uniformizer is $T^{1/2}$, the residue field is $k$, and the characteristic of both $k((T^{1/2}))$ and $k$ is $p$. Both fields have the same abstract residue data, yet one has characteristic $0$ and the other has characteristic $p$. The characteristic of $K$ is not determined by the residue field alone — it is an independent invariant that reflects how $p$ sits inside $\mathcal{O}_K$. [/example] The classification theorem for complete DVFs will treat these two cases very differently. For equal characteristic, the structure is controlled by the residue field alone, via the Teichmüller lift. For mixed characteristic, the structure requires the Witt vector construction. The obstruction to uniformity is precisely the one exhibited above: in equal characteristic $p$, the Frobenius on $\mathcal{O}_K$ commutes with addition, so the Teichmüller map is a ring section; in mixed characteristic $0$, the Frobenius does not commute with addition (since $p \neq 0$ in $\mathcal{O}_K$), so the Teichmüller map can only be a multiplicative section. ### Perfect Rings and the Frobenius The structure theorems for DVFs require the residue field to have a specific property. [definition: Perfect Ring] Let $R$ be a ring of characteristic $p > 0$. We say $R$ is **perfect** if the Frobenius map $\phi: R \to R$, $x \mapsto x^p$, is an automorphism of $R$, i.e., every element of $R$ has a unique $p$-th root. [/definition] [remark: Perfect Fields] A field $F$ of characteristic $p$ is perfect if and only if every finite extension of $F$ is separable. In particular, every finite field $\mathbb{F}_q$ (with $q = p^n$) is perfect: the Frobenius map $x \mapsto x^p$ is injective (fields have no zero divisors), and injectivity on a finite set implies surjectivity. [/remark] Perfection is the condition that lets us extract $p$-th roots of residue field elements, which is exactly what we need to construct canonical lifts from the residue field back to $\mathcal{O}_K$. ## Teichmüller Lifts To see why we need a special lifting procedure, consider $\mathbb{Z}_p$ with the naive choice of coset representatives $S = \{0, 1, \ldots, p-1\}$ for $\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{Z}/p\mathbb{Z}$. Every element of $\mathbb{Z}_p$ can be expanded as $\sum_{n=0}^\infty a_n p^n$ with $a_n \in S$. This works for additive purposes, but it respects essentially no multiplicative structure: for instance, the product of representatives is not a representative ($(p-1)^2 = p^2 - 2p + 1 \equiv 1$ in $\mathbb{Z}/p\mathbb{Z}$ but $(p-1)^2 = p^2 - 2p + 1 \notin S$ as an integer). Can we instead choose the coset representatives to form a multiplicative subgroup of $\mathcal{O}_K^\times$? The Teichmüller map provides exactly such a multiplicative section. The idea is iterative. Given $a \in k_K$, any naive lift $\alpha \in \mathcal{O}_K$ satisfies $[\alpha] \equiv a \pmod{\pi}$ but the choice of $\alpha$ is highly non-canonical. If instead we lift the $p$-th root $a^{p^{-1}} \in k_K$ (which exists since $k_K$ is perfect) to $\alpha_1 \in \mathcal{O}_K$ and then form $\alpha_1^p$, we get a lift of $a$ that is better determined: two choices of $\alpha_1$ differing by $x \in \pi \mathcal{O}_K$ produce values $\alpha_1^p$ differing by $O(\pi^2)$ (the cross terms involve $p \in \pi \mathcal{O}_K$ since $\operatorname{char}(k_K) = p$). Repeating the procedure — lifting $a^{p^{-n}}$ to $\alpha_n$ and taking $\alpha_n^{p^n}$ — produces a sequence that is Cauchy and whose limit is independent of all choices. The following key lemma makes this precise. [quotetheorem:2331] [citeproof:2331] The lemma captures the key cancellation mechanism: in characteristic $p$, raising to the $p$-th power is "almost a ring homomorphism" on $R/x^{k+1}R$ — the error from the previous level $R/x^k R$ is killed by the extra power of $x$. This improvement-of-approximation structure is exactly what drives the iterative construction of the Teichmüller lift. Note crucially that the argument uses $p \in xR$ (characteristic $p$ mod $x$); in characteristic $0$ one only has $p \in xR$ if $x \mid p$ in $R$, so for $R = \mathbb{Z}_p$ and $x = p$ the lemma does apply, but for $R = \mathbb{Z}_p$ and $x = p-1$ (say) it would not. The lemma is what makes the Frobenius iteration converge. With the lemma in hand, we can prove the Teichmüller theorem. [quotetheorem:2332] [definition: Teichmüller Map] The unique multiplicative section $[-]: R/xR \to R$ of the reduction map, as given by the theorem, is called the **Teichmüller map**. The element $[a]$ is called the **Teichmüller lift** or **Teichmüller representative** of $a$. [/definition] [citeproof:2332] [example: Teichmüller Representatives in $\mathbb{Z}_p$] Take $R = \mathbb{Z}_p$ and $x = p$. The Teichmüller map $[-]: \mathbb{F}_p \to \mathbb{Z}_p$ satisfies $[a]^{p-1} = [a^{p-1}] = [1] = 1$ for $a \neq 0$. So the Teichmüller representatives of nonzero elements of $\mathbb{F}_p$ are precisely the $(p-1)$-th roots of unity in $\mathbb{Z}_p$. These are the roots of $x^{p-1} - 1$ that we constructed via Hensel's lemma in Chapter 2. The Teichmüller map gives a canonical choice of coset representatives for $\mathbb{Z}_p / p\mathbb{Z}_p \cong \mathbb{F}_p$ that is multiplicative — unlike the naive set $\{0, 1, \ldots, p-1\}$. [/example] The Teichmüller representatives are useful for theoretical purposes (they make canonical sense as ring-theoretic objects), but one should not expect them to be computationally tractable. From the categorical perspective, the Teichmüller map is the unit of an adjunction: the forgetful functor from perfect rings of characteristic $p$ to sets has a left adjoint (free perfection), and the Teichmüller map is the canonical section arising from that adjunction. This is why the map is unique: it is the only multiplicative section compatible with the ring structure of the residue field, and any two such sections must agree on $p$-power roots, hence everywhere by the density of the iteration. ## Structure of Complete DVFs of Equal Characteristic Can we classify all complete DVFs of equal characteristic? The answer is yes, at least when the residue field is perfect, and the classification is essentially tautological once the Teichmüller map is in hand: every complete DVF of equal characteristic $p$ with perfect residue field is a formal Laurent series field over that residue field. The Teichmüller map provides exactly the ring-theoretic section needed to identify the coefficients. The Teichmüller map immediately yields a classification of complete DVFs of equal characteristic. [quotetheorem:2380] [citeproof:2380] [remark: Classification of Equal Characteristic Local Fields] Let $K$ be a local field of equal characteristic $p$. Then $k_K \cong \mathbb{F}_q$ for some power $q = p^f$ of $p$ (since $k_K$ is finite). Every finite field is perfect. Applying the theorem, $K \cong \mathbb{F}_q((T))$. So the equal characteristic local fields are exactly $\{\mathbb{F}_q((T)) : q = p^f, p \text{ prime}\}$. [/remark] ## Witt Vectors and Mixed Characteristic Structure The mixed characteristic analogue is more subtle: we cannot use the Teichmüller map alone to reconstruct $\mathcal{O}_K$ from $k_K$, because the ring structure of $\mathcal{O}_K$ is not determined by $k_K$ alone — it depends on how the characteristic $p$ element lifts. The correct construction is the ring of **Witt vectors**, which is the universal mixed-characteristic DVR with a given perfect residue field. We include a streamlined treatment of Witt vectors as essential context for the classification of mixed characteristic DVFs — without it, the relationship between $\mathbb{Q}_p$ and $\mathbb{F}_p((T))$ would remain mysterious. ### Strict $p$-Rings The equal characteristic structure theorem used the fact that $\operatorname{char}(\mathcal{O}_K) = p$ to make the Teichmüller map additive. In the mixed characteristic case, $\mathcal{O}_K$ has characteristic $0$, so the Teichmüller map is only multiplicative. To reconstruct $\mathcal{O}_K$ from the residue field $k_K$, we need a ring in which: 1. Addition is well-defined and compatible with the Teichmüller map, 2. The ring is $p$-adically complete (so that $\pi$-adic series converge), 3. The residue ring mod $p$ is the given perfect field $k_K$. The Witt vector construction builds exactly such a ring, universally. The precise axioms needed are captured by the notion of a strict $p$-ring. [definition: Strict $p$-Ring] A ring $A$ is called a **strict $p$-ring** if it is $p$-torsion free, $p$-adically complete, and $A/pA$ is a perfect ring of characteristic $p$. [/definition] The conditions in the definition are precisely those needed for the Teichmüller map $[-]: A/pA \to A$ to exist (with $x = p$), and for the Witt vector decomposition to work. [example: $\mathbb{Z}_p$ as a Strict $p$-Ring] $\mathbb{Z}_p$ is a strict $p$-ring: it is $p$-torsion free (as a subring of $\mathbb{Q}$), $p$-adically complete by construction, and $\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{F}_p$ is perfect. This example motivates the entire construction. [/example] The defining feature of a strict $p$-ring is its Witt vector expansion. The conditions in the definition ensure two things: $p$-torsion freeness guarantees that $p$ is not a zero divisor (so we can cancel $p$ in arguments), and $p$-adic completeness ensures that the infinite series $\sum [a_n] p^n$ converge. Together with perfection of $A/pA$, these are exactly the conditions needed for the following unique expansion theorem — the mixed-characteristic analogue of the $\pi$-adic expansion. [quotetheorem:2333] [citeproof:2333] The expansion $\sum [a_n] p^n$ is the exact analogue of the $\pi$-adic expansion in the equal characteristic case, with Teichmüller representatives as digits. The key difference: in the equal characteristic case the digit ring $k_K$ embeds into $\mathcal{O}_K$ as a subring (the Teichmüller map is a ring homomorphism), while in the mixed characteristic case $k_K$ does not embed as a ring — the Teichmüller map is only multiplicative, so addition of digits does not correspond to addition in $\mathcal{O}_K$. A pitfall: one might think the expansion $\sum [a_n] p^n$ defines a ring structure on $k_K^{\mathbb{N}}$ by "carry-free" addition. This fails because $(p-1) + 1 = p$ in $\mathbb{Z}_p$ corresponds to $[p-1] + [1] = p \cdot 1 = [0] \cdot p^1$, which is not $[0]$ in position $0$ without a carry. The Witt vector construction is precisely the machine that encodes these carries systematically using explicit polynomial formulas. ### Lifts of Homomorphisms A key feature of strict $p$-rings is that homomorphisms between their residue rings lift uniquely. This is the "functorial" heart of the Witt vector construction: it says that the assignment $R \mapsto W(R)$ (residue ring to strict $p$-ring) is not just an existence result but a full equivalence of categories, with morphisms on both sides in bijection. The proof is a direct application of the Witt expansion, using the fact that any ring homomorphism must send Teichmüller lifts to Teichmüller lifts. [quotetheorem:2334] [citeproof:2334] Perfection of the residue rings is essential for the lifting to work: the uniqueness argument requires that $\psi([a]) = [f(a)]$, which in turn requires extracting $p$-th roots $a^{p^{-n}}$ in $A/pA$ and $f(a)^{p^{-n}}$ in $B/pB$. Without perfection, these roots need not exist, and the Teichmüller iterates that pin down the lift would not converge to a well-defined element. Categorically, the lifting theorem says that the functor $W$ from perfect $\mathbb{F}_p$-algebras to strict $p$-rings is fully faithful: it is a bijection on morphism sets, not just on objects. A concrete instance is the Frobenius automorphism $\phi: x \mapsto x^p$ on $\mathbb{F}_p$, which is the identity; its unique lift to $\mathbb{Z}_p$ is the identity map $\mathbb{Z}_p \to \mathbb{Z}_p$. More interestingly, for $\mathbb{F}_{p^f}$ the Frobenius $\phi: x \mapsto x^p$ is a non-trivial automorphism of order $f$, and it lifts to the unique automorphism of $W(\mathbb{F}_{p^f})$ that acts as Frobenius on the residue field — this is exactly the arithmetic Frobenius that generates $\operatorname{Gal}(W(\mathbb{F}_{p^f})[p^{-1}] / \mathbb{Q}_p)$. More generally, the lifting property holds when $B$ is any ring satisfying the Teichmüller hypotheses: if $B$ is $x$-adically complete and $B/xB$ is perfect of characteristic $p$, and $f: A/pA \to B/xB$ is a ring homomorphism, then $f$ lifts uniquely to $F: A \to B$. ### Existence and Uniqueness of Witt Vectors The lifting theorem shows that if a strict $p$-ring $W(R)$ with $W(R)/pW(R) \cong R$ exists, it is unique. The existence half requires building $W(R)$ explicitly from $R$. The construction is a free-then-quotient argument: first build the free strict $p$-ring on the generators of $R$ (by freely adjoining $p$-power roots and completing), then quotient by the kernel of the surjection to $R$. The universal property (functoriality) makes $W$ a functor from perfect rings of characteristic $p$ to strict $p$-rings, and the lifting theorem shows this functor is an equivalence of categories. The perfectness of $R$ is essential: without it the Teichmüller map would not exist, the Witt expansion would break down, and the lift of a homomorphism $f: R \to R'$ would not be unique. [quotetheorem:2335] [citeproof:2335] [example: Witt Vectors of Finite Fields] The most important case: $W(\mathbb{F}_p) = \mathbb{Z}_p$. This follows from the uniqueness of $W(\mathbb{F}_p)$ and the fact that $\mathbb{Z}_p$ is a strict $p$-ring with $\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{F}_p$. More generally, $W(\mathbb{F}_q)$ is the unique unramified extension of $\mathbb{Z}_p$ of degree $f$ (where $q = p^f$), which we will identify more precisely in the ramification theory of Chapter 5. [/example] The Witt vector theorem gives us a functor $W$ from perfect rings of characteristic $p$ to strict $p$-rings. By the lifting theorem, this functor is an equivalence of categories: every morphism $f: R \to R'$ lifts uniquely to $F: W(R) \to W(R')$, and every morphism between strict $p$-rings is of this form. Perfectness of $R$ is what buys both existence and uniqueness of the lift: surjectivity of Frobenius ensures we can always find $p$-th roots to carry out the iterative construction, and injectivity ensures those $p$-th roots are unique, making the Teichmüller map well-defined. If $R$ were not perfect — say $R = \mathbb{F}_p[T]$, which has no $p$-th root of $T$ — then $W(R)$ would still exist as a $p$-torsion-free complete ring, but the residue ring $W(R)/pW(R) \cong R$ would not be perfect, breaking the lifting bijection. ### Structure of Mixed Characteristic Complete DVRs With the Witt vector machine in hand, we can now describe all mixed characteristic complete DVRs. The central distinction is whether the prime $p$ is itself the uniformizer (the unramified case, $e = 1$) or whether $p$ is a higher power of the uniformizer (the ramified case, $e > 1$). In the unramified case the structure is completely pinned down by the Witt vector theorem: $\mathcal{O}_K \cong W(k_K)$. In the ramified case, $p$ becomes an Eisenstein element — it satisfies a polynomial $p = \pi^e + \text{lower terms}$ — and the structure is a finite extension of $W(k_K)$. The theorem below makes the ramified case precise. [quotetheorem:2336] [citeproof:2336] [definition: Absolute Ramification Index] Let $A$ be a DVR of mixed characteristic $p$ with normalized valuation $v_A$. The integer $e = v_A(p)$ is called the **absolute ramification index** of $A$. [/definition] When $e = 1$, the prime $p$ is itself the uniformizer, and the structure is completely determined by the residue field. [quotetheorem:2337] [citeproof:2337] When $e > 1$, the theorem fails: $A$ is no longer a strict $p$-ring because $p$ is not a uniformizer — rather $p = u\pi^e$ for some unit $u$, and $A/pA$ is not a field but has nilpotent elements (the image of $\pi$ satisfies $\bar{\pi}^e = 0$). Geometrically, the unramified case $e = 1$ is the simplest possible structure because the residue field completely determines the DVR: knowing $k$ pins down $A \cong W(k)$ with no further choices. This is the precise sense in which "unramified means the residue field determines everything" — once one fixes the residue field $k$, there is exactly one unramified complete DVR with that residue field, while the ramified case introduces additional data (the Eisenstein polynomial of the uniformizer over $W(k)$). The general mixed characteristic case allows $e \geq 1$, and the structure is described as a finite extension of $W(k)$. [quotetheorem:2338] [citeproof:2338] The most important consequence is the classification of mixed characteristic local fields. [quotetheorem:2339] [citeproof:2339] Together with the equal characteristic classification, we obtain the complete picture: every local field is either $\mathbb{F}_q((T))$ for some finite field $\mathbb{F}_q$, or a finite extension of $\mathbb{Q}_p$ for some prime $p$. The Witt vector ring $W(\mathbb{F}_q)$ is the "maximally unramified" mixed characteristic DVR with residue field $\mathbb{F}_q$; any mixed characteristic complete DVR with residue field $\mathbb{F}_q$ is a finite extension of $W(\mathbb{F}_q)$, classified by its uniformizer and the minimal polynomial of that uniformizer over $W(\mathbb{F}_q)$. This chapter's structure theorems form the foundation for the ramification theory developed in Chapter 5, where we classify finite extensions of local fields according to how the prime $p$ factors in the extension — the key invariants being the ramification index $e$ and the inertia degree $f$ whose product equals the degree of the extension. Valued fields exhibit diverse topologies depending on how their valuations behave, but the most tractable—and most important—are those whose valuations take discrete values in Z. We now specialise to discretely valued fields, where the interplay between ramification index and residue degree becomes both computable and central. # 3. Discretely Valued Fields This chapter steps back from the algebraic structure theory of complete discretely valued fields developed in Chapter 3 to explore analysis on $\mathbb{Z}_p$. The central question is: what does it mean for a function $\mathbb{Z}_p \to \mathbb{Q}_p$ to be continuous, and how can such functions be represented explicitly? The answer — Mahler's theorem — is a $p$-adic analogue of the classical Weierstrass approximation theorem, replacing polynomials by a basis of binomial coefficients. We also briefly examine the $p$-adic exponential and logarithm, which converge not on all of $\mathbb{Q}_p$ but only on discs determined by $p$. ## The $p$-adic Exponential and Logarithm In real analysis, every analytic function can be expressed as a power series converging on some interval. In the $p$-adic world, power series still converge, but the domains of convergence are controlled by the $p$-adic absolute value and the arithmetic of factorials and denominators — not by real distance from the origin. We define the exponential and logarithm by the same formulas as in the real and complex cases: \begin{align*} \exp(x) &= \sum_{n=0}^{\infty} \frac{x^n}{n!}, \quad \log(1 + x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}. \end{align*} The question is: for which $x \in \mathbb{Q}_p$ (or a complete extension) do these series converge? [quotetheorem:2340] [citeproof:2340] [remark: Why the radius $p^{-1/(p-1)}$] The threshold $p^{-1/(p-1)}$ arises because the $p$-adic valuation of $n!$ grows like $n/(p-1)$, a consequence of the many factors of $p$ accumulated in the factorial. The exponential series converges in a strictly smaller disc than the logarithm series — this asymmetry reflects the presence of $n!$ in the denominator versus the much simpler $n$. [/remark] To see these constraints in practice, let us compute the convergence radii for a specific prime and check which elements lie inside. [example: Convergence in $\mathbb{Q}_p$] For $p = 5$, the exponential series converges for $|x|_5 < 5^{-1/4}$. The element $5 \in \mathbb{Q}_5$ satisfies $|5|_5 = 5^{-1}$, and since $5^{-1} < 5^{-1/4}$, we have $\exp(5) \in \mathbb{Z}_5$. On the other hand, for $p = 2$, the threshold is $|x|_2 < 2^{-1}$, so $\exp(2)$ does not converge in $\mathbb{Q}_2$ because $|2|_2 = 2^{-1}$ is not strictly less than $2^{-1}$. [/example] ## The Space of Continuous Functions $C(\mathbb{Z}_p, \mathbb{Q}_p)$ In real analysis, the Weierstrass theorem tells us polynomials approximate continuous functions. In the $p$-adic world, the compact space $\mathbb{Z}_p$ has a richer structure. What does a basis for $C(\mathbb{Z}_p, \mathbb{Q}_p)$ look like, and why are polynomials not the right answer? We now turn to the main subject: continuous functions from $\mathbb{Z}_p$ to $\mathbb{Q}_p$. Recall that $\mathbb{Z}_p$ is compact (it is the valuation ring of the complete field $\mathbb{Q}_p$, and equals the inverse limit $\varprojlim \mathbb{Z}/p^n\mathbb{Z}$, which is a closed and bounded subset of $\mathbb{Q}_p$). Because $\mathbb{Z}_p$ is compact, every continuous function $f: \mathbb{Z}_p \to \mathbb{Q}_p$ is bounded, and the supremum norm \begin{align*} \|f\| = \sup_{x \in \mathbb{Z}_p} |f(x)|_p \end{align*} is finite and is actually attained. [definition: The space $C(\mathbb{Z}_p, \mathbb{Q}_p)$] We write $C(\mathbb{Z}_p, \mathbb{Q}_p)$ for the $\mathbb{Q}_p$-vector space of all continuous functions $f: \mathbb{Z}_p \to \mathbb{Q}_p$, equipped with the supremum norm $\|f\| = \sup_{x \in \mathbb{Z}_p} |f(x)|_p$. [/definition] This is a natural setting for analysis: we have a vector space of functions with a well-behaved norm. The next step is to verify that this norm space is complete, making it a Banach space. [quotetheorem:2341] The non-archimedean property of the norm follows directly from the strong triangle inequality: $\|f + g\| \leq \max(\|f\|, \|g\|)$. Completeness is a standard result: a uniform limit of continuous functions is continuous, and in the non-archimedean setting it suffices for $\|f_n - f_m\| \to 0$ for the sequence to converge uniformly. ### Binomial Coefficient Functions In classical analysis, the Weierstrass approximation theorem says that polynomials are dense in $C([0,1], \mathbb{R})$. The basis $\{1, x, x^2, \ldots\}$ is natural there. For $C(\mathbb{Z}_p, \mathbb{Q}_p)$, the correct basis turns out to be the binomial coefficient functions. For $n \geq 1$, define the polynomial \begin{align*} \binom{x}{n} = \frac{x(x-1)(x-2)\cdots(x-n+1)}{n!}, \end{align*} and set $\binom{x}{0} = 1$ for all $x \in \mathbb{Z}_p$. Since this is a polynomial in $x$, it defines a continuous function $\mathbb{Z}_p \to \mathbb{Q}_p$. Why does this function take values in $\mathbb{Z}_p$, not merely in $\mathbb{Q}_p$? For $x \in \mathbb{Z}_{\geq 0}$, the value $\binom{x}{n}$ is a non-negative integer, so $\binom{x}{n} \in \mathbb{Z} \subset \mathbb{Z}_p$. Since $\mathbb{Z}_{\geq 0}$ is dense in $\mathbb{Z}_p$ (the non-negative integers are $p$-adically dense in $\mathbb{Z}_p$, as shown in the basic theory of the $p$-adic numbers), continuity forces $\binom{x}{n} \in \mathbb{Z}_p$ for all $x \in \mathbb{Z}_p$. [remark: Integer-valuedness] The key point is that $\left|\binom{x}{n}\right|_p \leq 1$ for all $x \in \mathbb{Z}_p$ and all $n \geq 0$. This means the binomial functions are "bounded by $1$" in the supremum norm, which is essential for the convergence of Mahler series. [/remark] ## The Difference Operator $\Delta$ Classical analysis relies on differentiation to study functions. But on $\mathbb{Z}_p$, every locally constant function has derivative zero everywhere — differentiation sees almost nothing. What discrete analogue can we use instead? To see the problem concretely: the function $f: \mathbb{Z}_p \to \mathbb{Q}_p$ defined by $f(x) = 1$ if $|x|_p < 1$ (i.e. $x \in p\mathbb{Z}_p$) and $f(x) = 0$ otherwise is continuous and locally constant but non-constant. Its classical derivative is $\lim_{h \to 0} (f(x+h) - f(x))/h$. For $x \notin p\mathbb{Z}_p$ and $|h|_p$ small enough, $x + h \notin p\mathbb{Z}_p$ either, so $f(x+h) - f(x) = 0$ and the derivative is $0$. Similarly for $x \in p\mathbb{Z}_p$. Locally constant functions form a dense subspace of $C(\mathbb{Z}_p, \mathbb{Q}_p)$, yet they all have derivative $0$. Differentiation is useless here. We need a discrete substitute that can actually distinguish these functions. Before stating Mahler's theorem, we introduce the forward difference operator, which plays this role. [definition: Forward Difference Operator] The forward difference operator $\Delta: C(\mathbb{Z}_p, \mathbb{Q}_p) \to C(\mathbb{Z}_p, \mathbb{Q}_p)$ is defined by \begin{align*} (\Delta f)(x) = f(x+1) - f(x). \end{align*} [/definition] The operator $\Delta$ is linear and, by the non-archimedean triangle inequality, satisfies $\|\Delta f\| \leq \|f\|$, so $\|\Delta\| \leq 1$. The iterates $\Delta^n$ can be expressed explicitly by the binomial formula: induction gives \begin{align*} \Delta^n f(x) = \sum_{i=0}^{n} (-1)^i \binom{n}{i} f(x + n - i). \end{align*} This formula makes $\Delta$ into the discrete analogue of differentiation. The binomial coefficient functions are the natural eigenfunctions of $\Delta$: a short computation using the Pascal identity $\binom{x+1}{n} - \binom{x}{n} = \binom{x}{n-1}$ gives $\Delta \binom{x}{n} = \binom{x}{n-1}$. In particular, $\Delta$ shifts the index of a binomial coefficient function down by one. [example: Action of $\Delta$ on binomials] Let us verify $\Delta\binom{x}{2} = \binom{x}{1}$. We compute directly: \begin{align*} \Delta\binom{x}{2} &= \binom{x+1}{2} - \binom{x}{2} = \frac{(x+1)x}{2} - \frac{x(x-1)}{2} = \frac{x^2 + x - x^2 + x}{2} = x = \binom{x}{1}. \end{align*} This reflects the general identity $\binom{x}{n} + \binom{x}{n-1} = \binom{x+1}{n}$, which holds for all $x \in \mathbb{Z}_p$ by density (it holds for $x \in \mathbb{Z}_{\geq n}$, and both sides are continuous functions agreeing on an infinite set, hence equal everywhere). [/example] ## Mahler Coefficients and Their Decay Given a continuous function $f$ on $\mathbb{Z}_p$, we want to extract coefficients that determine $f$ uniquely. But extracting these coefficients requires showing they decay — what controls the size of $\Delta^n f(0)$? Given $f \in C(\mathbb{Z}_p, \mathbb{Q}_p)$, we define the Mahler coefficients by evaluating the iterated difference operator at $0$. [definition: Mahler Coefficient] Let $f \in C(\mathbb{Z}_p, \mathbb{Q}_p)$. The $n$th Mahler coefficient of $f$ is \begin{align*} a_n(f) = \Delta^n f(0) = \sum_{i=0}^{n} (-1)^i \binom{n}{i} f(n-i). \end{align*} [/definition] Note that $a_0(f) = f(0)$ and $a_1(f) = f(1) - f(0)$. The formula says that $a_n(f)$ is a finite alternating sum of values of $f$ at the non-negative integers $0, 1, \ldots, n$. For Mahler's theorem to hold, we need $a_n(f) \to 0$ as $n \to \infty$. Since $\|\Delta\| \leq 1$, the sequence $\|\Delta^n f\|$ is non-increasing, so the coefficients do not blow up. The real work is showing they eventually decay. The key tool is a lemma that uses the compact structure of $\mathbb{Z}_p$ — specifically uniform continuity. [quotetheorem:2381] [citeproof:2381] The contraction factor $1/p$ here is not arbitrary: it is exactly the gap between consecutive values of $|\cdot|_p$. The argument uses uniform continuity in an essential way — it would fail for a merely pointwise-continuous function on a non-compact domain, since we need a single modulus of continuity that works for all $x$ simultaneously. The iterate $\Delta^{p^k}$ (rather than $\Delta^k$) is needed because the binomial coefficients $\binom{p^k}{i}$ for $0 < i < p^k$ are divisible by $p$, which is what forces the $p$-adic reduction. [quotetheorem:2342] [citeproof:2342] What this theorem says at the level of Banach space theory is that the Mahler coefficient map $f \mapsto (a_n(f))_{n \geq 0}$ is an isometric embedding of $C(\mathbb{Z}_p, \mathbb{Q}_p)$ into $c_0$. Injectivity means two different continuous functions cannot share the same Mahler coefficients — the sequence $(a_n(f))$ determines $f$ completely. The norm bound $\|(a_n(f))\|_{c_0} \leq \|f\|$ means this embedding does not inflate norms. We do not yet know it is surjective (that every sequence in $c_0$ arises from some continuous function), but injectivity and norm-decrease are already powerful: they tell us $C(\mathbb{Z}_p, \mathbb{Q}_p)$ embeds isometrically as a closed subspace of $c_0$. ## Mahler's Theorem We have shown that every continuous $f$ produces Mahler coefficients tending to zero. The central question remains: does the Mahler series $\sum_{n \geq 0} a_n(f) \binom{x}{n}$ actually converge back to $f$? Before tackling this, it is worth understanding why the naive approach — using monomials $x^n$ as a basis — fails. The functions $x^n$ are continuous on $\mathbb{Z}_p$ but do not form a good basis. There is no non-archimedean analogue of the estimate $|x^n/n!|_p \leq 1$ for $x \in \mathbb{Z}_p$: indeed $\|x^n\| = 1$ for all $n \geq 1$ (since $|1^n|_p = 1$), but the sequence $(1, 1, 1, \ldots)$ does not lie in $c_0$. More concretely, if we tried to expand $f$ in the monomial basis, the "coefficients" extracted by divided differences would have no reason to decay. The binomial coefficient functions $\binom{x}{n}$ are different: they satisfy $\left|\binom{x}{n}\right|_p \leq 1$ for all $x \in \mathbb{Z}_p$ (because $\binom{x}{n} \in \mathbb{Z}_p$), so they are uniformly bounded in $p$-adic norm, and $\Delta \binom{x}{n} = \binom{x}{n-1}$ ensures the coefficient extraction process works cleanly. We now have enough structure to state and prove the main theorem of this chapter. Let $c_0$ denote the Banach space of sequences $(a_n)_{n \geq 0}$ in $\mathbb{Q}_p$ with $a_n \to 0$, normed by $\|(a_n)\| = \max_{n \geq 0} |a_n|_p$ (the maximum exists because $|a_n|_p \to 0$, so the supremum is attained). [quotetheorem:2343] The key to the proof is constructing a pair of inverse isometric isomorphisms between $C(\mathbb{Z}_p, \mathbb{Q}_p)$ and $c_0$, and verifying that they are indeed inverse to each other via a purely formal argument. We first define the "reconstruction" map. [quotetheorem:2382] [citeproof:2382] We now have two maps: \begin{align*} F: C(\mathbb{Z}_p, \mathbb{Q}_p) &\to c_0, & F(f) &= (a_n(f))_{n \geq 0}, \\ G: c_0 &\to C(\mathbb{Z}_p, \mathbb{Q}_p), & G(a) &= f_a. \end{align*} We know $F$ is injective and norm-decreasing (from the theorem on Mahler coefficients), and $G$ is norm-decreasing with $F \circ G = \mathrm{id}_{c_0}$ (from the reconstruction theorem). The following purely formal lemma finishes the proof of Mahler's theorem. [quotetheorem:2344] [citeproof:2344] This lemma is a clean and reusable tool in non-archimedean functional analysis: whenever you can exhibit two norm-decreasing maps between Banach spaces that compose to the identity in one direction, and one of which is injective, you get an isometric isomorphism for free. The argument makes no use of $p$-adic structure — it applies in any normed space setting. The key insight is that injectivity and norm-decrease together force the maps to preserve norms, so the compression one direction guarantees expansion in the other is impossible. Putting everything together: $G \circ F = \mathrm{id}$ and $F \circ G = \mathrm{id}$, so $F$ and $G$ are mutually inverse isometric isomorphisms. In particular, every $f \in C(\mathbb{Z}_p, \mathbb{Q}_p)$ equals $G(F(f)) = f_{(a_n(f))}$, which is precisely the Mahler expansion. The norm equality $\|f\| = \|(a_n(f))\|_{c_0} = \max_n |a_n(f)|_p$ follows from $F$ being isometric. This completes the proof of Mahler's theorem. [explanation: What Mahler's theorem really says] Mahler's theorem is the $p$-adic analogue of the Weierstrass approximation theorem, but it is in fact stronger: it gives an exact representation (not merely dense approximation) of every continuous function. The classical Weierstrass theorem says that polynomials are dense in $C([0,1], \mathbb{R})$, but not every continuous function is a polynomial. By contrast, Mahler's theorem says that every continuous $f: \mathbb{Z}_p \to \mathbb{Q}_p$ is exactly represented by its Mahler series $\sum a_n \binom{x}{n}$, with no error term. The space $c_0$ of sequences tending to zero is the natural coefficient space. The condition $a_n \to 0$ is both necessary (for the series to converge) and sufficient (as the theorem shows). The isometry $C(\mathbb{Z}_p, \mathbb{Q}_p) \cong c_0$ also makes concrete the "size" of the space of continuous functions: a function is determined exactly by how its alternating differences at $0$ decay. Why binomial coefficients and not monomials $x^n$? The monomials do not have any special non-archimedean properties. The binomial functions $\binom{x}{n}$ are distinguished by two facts: they satisfy $\left|\binom{x}{n}\right|_p \leq 1$ for all $x \in \mathbb{Z}_p$ (integer-valuedness), and they diagonalise $\Delta$ in the sense that $\Delta \binom{x}{n} = \binom{x}{n-1}$. These two properties together make the Mahler expansion work. [/explanation] To make Mahler's theorem concrete, let us compute the Mahler expansion of an explicit example and verify the coefficient decay directly. [example: Mahler expansion of a locally constant function] Let $f: \mathbb{Z}_p \to \mathbb{Q}_p$ be the characteristic function of $p\mathbb{Z}_p$, i.e. $f(x) = 1$ if $p \mid x$ and $f(x) = 0$ otherwise. This is locally constant, hence continuous. The Mahler coefficients are $a_n(f) = \sum_{i=0}^{n} (-1)^i \binom{n}{i} f(n - i)$. Here $f(n-i) = 1$ if $p \mid (n-i)$ and $0$ otherwise, so \begin{align*} a_n(f) = \sum_{\substack{i=0 \\ p \mid (n-i)}}^{n} (-1)^i \binom{n}{i}. \end{align*} For $0 \leq n < p$: the only multiple of $p$ in $\{n, n-1, \ldots, 0\}$ is $0$ itself (since $0 < n < p$ means none of $1, \ldots, n$ is divisible by $p$). Thus only $i = n$ contributes, giving $a_n(f) = (-1)^n \binom{n}{n} = (-1)^n$. In particular $|a_n(f)|_p = 1$ for $0 \leq n < p$. For $n = p$: the multiples of $p$ in $\{p, p-1, \ldots, 0\}$ are $p$ and $0$. The term $i = 0$ contributes $f(p) = 1$, and the term $i = p$ contributes $(-1)^p f(0) = (-1)^p$. All other values $n - i \in \{1, \ldots, p-1\}$ are not divisible by $p$. Thus \begin{align*} a_p(f) = \binom{p}{0} + (-1)^p \binom{p}{p} = 1 + (-1)^p = 1 + (-1)^p. \end{align*} For odd $p$ this is $0$, so $|a_p(f)|_p = 0 < 1$. For $p = 2$ this gives $2$, so $|a_2(f)|_2 = |2|_2 = 1/2 < 1$. In both cases, $|a_p(f)|_p < 1$, confirming the decay has begun. More generally, for $n \geq p$, the sum $a_n(f) = \sum_{j : p \mid j, 0 \leq n-j \leq n} (-1)^{n-j} \binom{n}{n-j}$ is a sum over multiples of $p$ in $\{0, 1, \ldots, n\}$. Kummer's theorem shows that $v_p(a_n(f)) \to \infty$ as $n \to \infty$, so $|a_n(f)|_p \to 0$, as required by Mahler's theorem. For instance, $a_{p^2}(f) = \sum_{j=0}^{p} (-1)^{p^2 - jp} \binom{p^2}{jp}$; by Lucas' theorem, $\binom{p^2}{jp} \equiv \binom{p}{j} \pmod{p}$, so modulo $p$ this sum is $\sum_{j=0}^{p} (-1)^{jp} \binom{p}{j} = (1 - (-1)^p)^p$, which is $0 \pmod{p}$ for odd $p$. Hence $|a_{p^2}(f)|_p \leq 1/p$. [/example] The algebraic classification of extensions by ramification and residue degree is complete, yet it reveals nothing about the analytical structure of Z_p and its extensions. This chapter pivots to analysis, exploring convergence, continuous functions, and measure on p-adic fields. # 4. Some p-adic Analysis The structure theorems of Chapter 3 told us what local fields look like in isolation: they are either $\mathbb{F}_q((T))$ or finite extensions of $\mathbb{Q}_p$. But the richest phenomena in local number theory emerge when we pass to extensions — and extensions of local fields are not all alike. Some extensions leave the uniformizer essentially unchanged; others force the uniformizer to split into finer pieces. Ramification theory is the study of exactly this distinction. The central invariants are the ramification index $e$ and the inertia degree $f$, which measure how much of the extension is "visible" in the residue field and how much is "invisible." The fundamental identity $[L:K] = ef$ ties these invariants to the degree of the extension and is the cornerstone of the whole theory. Throughout this chapter, $p$ denotes the characteristic of the residue field of any local field under consideration. ## The Fundamental Invariants: Ramification Index and Inertia Degree Let $L/K$ be a finite extension of local fields. Since $\mathcal{O}_K \subseteq \mathcal{O}_L$ and $\mathfrak{m}_K \subseteq \mathfrak{m}_L$, there is a natural injection \begin{align*} k_K = \frac{\mathcal{O}_K}{\mathfrak{m}_K} \hookrightarrow \frac{\mathcal{O}_L}{\mathfrak{m}_L} = k_L. \end{align*} So every extension of local fields induces an extension of residue fields $k_L/k_K$. The question is: how does the structure of $L/K$ split between what happens at the residue level and what happens "above" the residue field? Two integers capture this decomposition. [definition: Inertia Degree] Let $L/K$ be a finite extension of local fields. The **inertia degree** of $L/K$ is \begin{align*} f_{L/K} = [k_L : k_K]. \end{align*} [/definition] The inertia degree measures the "horizontal" growth of the extension — how much the residue field expands. But extensions can also grow "vertically", by forcing the uniformizer of $K$ to split into finer pieces in $L$. A second invariant captures this. [definition: Ramification Index] Let $L/K$ be a finite extension of local fields, let $v_L$ be the normalized valuation of $L$, and let $\pi_K$ be a uniformizer of $K$. The **ramification index** of $L/K$ is \begin{align*} e_{L/K} = v_L(\pi_K). \end{align*} [/definition] The ramification index measures how the prime of $K$ splits in $L$: in the ring of integers, $\pi_K \mathcal{O}_L = \mathfrak{m}_L^{e_{L/K}}$, so $e_{L/K}$ counts the multiplicity with which the maximal ideal of $K$ becomes a power of the maximal ideal of $L$. When $e_{L/K} = 1$, the prime of $K$ is still a prime of $L$; when $e_{L/K} > 1$, the prime has "ramified." The inertia degree $f_{L/K}$ measures how much the residue field has grown. [example: Ramification and Inertia in $\mathbb{Q}_p$-Extensions] Consider $K = \mathbb{Q}_p$ and two extensions. First, let $L = \mathbb{Q}_p(\zeta)$ where $\zeta$ is a primitive $(p-1)$-th root of unity (if $p > 2$). The residue field of $\mathbb{Q}_p$ is $\mathbb{F}_p$, which already contains all $(p-1)$-th roots of unity (by Fermat's little theorem). So $\zeta$ reduces to a $(p-1)$-th root of unity in $\mathbb{F}_p$, and in fact $\mathbb{Q}_p$ already contains $\zeta$ by Hensel's lemma (the polynomial $x^{p-1} - 1$ splits completely over $\mathbb{F}_p$ and Hensel lifts the roots). This tells us nothing new from a ramification standpoint. Now consider $L = \mathbb{Q}_p(\pi_L)$ where $\pi_L$ is a root of $x^2 - p$. Then $v_L(\pi_L) = 1$ by definition, while $v_L(p) = v_L(\pi_L^2) = 2$. So $e_{L/\mathbb{Q}_p} = v_L(p) = 2$. The residue field of $L$ is still $\mathbb{F}_p$ (the uniformizer $\pi_L$ reduces to $0$ in $k_L$, and all units reduce to $\mathbb{F}_p$), so $f_{L/\mathbb{Q}_p} = 1$. We have $e_{L/\mathbb{Q}_p} f_{L/\mathbb{Q}_p} = 2 \cdot 1 = 2 = [L:\mathbb{Q}_p]$, as predicted by the fundamental identity. [/example] ## The Integral Structure of a Finite Extension Given a finite extension $L/K$, we want to describe $\mathcal{O}_L$ as an $\mathcal{O}_K$-module. Can we use a $K$-basis of $L$? What goes wrong? The short answer is that a $K$-basis for $L$ need not be an $\mathcal{O}_K$-basis for $\mathcal{O}_L$: rescaling by elements of $K$ cannot always turn an arbitrary basis into one that generates exactly $\mathcal{O}_L$. Nevertheless, the structure is as clean as possible: $\mathcal{O}_L$ is a free $\mathcal{O}_K$-module of rank $n = [L:K]$. [quotetheorem:2345] [citeproof:2345] This theorem is worth pausing over. It is not possible to simply scale an arbitrary $K$-basis of $L$ to get an $\mathcal{O}_K$-basis of $\mathcal{O}_L$: the structure of $\mathcal{O}_L$ can be finer than any rescaled $K$-basis. [example: Why Scaling a Basis Can Fail] Consider $\mathbb{Q}_2(\sqrt{2})/\mathbb{Q}_2$, which has $K$-basis $\{1, \sqrt{2}\}$. Since $|\sqrt{2}|_2 = 2^{-1/2}$, this element has absolute value less than $1$, so $\sqrt{2} \in \mathcal{O}_L$. To scale to absolute value $1$ we would multiply by $2^{1/2} = \sqrt{2}$ itself, which is circular: there is no element of $\mathbb{Q}_2$ that scales $\sqrt{2}$ to a unit. The correct $\mathcal{O}_K$-basis of $\mathcal{O}_L = \mathbb{Z}_2[\sqrt{2}]$ is simply $\{1, \sqrt{2}\}$. A subtler obstruction arises for $\mathbb{Q}_3(\sqrt{-1})/\mathbb{Q}_3$: the elements $1$ and $1 + 3\sqrt{-1}$ both have absolute value $1$, so they look like a candidate $\mathcal{O}_K$-basis. But $\sqrt{-1} \notin \mathbb{Z}_3 \cdot 1 + \mathbb{Z}_3 \cdot (1 + 3\sqrt{-1})$ — to express $\sqrt{-1}$, one would need $\frac{1}{3}(1 + 3\sqrt{-1} - 1) = \sqrt{-1}/1$, which requires dividing by $3$, not allowed in $\mathbb{Z}_3$. So this is not an $\mathcal{O}_K$-basis of $\mathcal{O}_L$. The existence of a free basis is guaranteed by the theorem, but finding it requires more care. [/example] ## The Fundamental Identity [quotetheorem:2398] [citeproof:2398] [remark: The Monogenic Structure] The conclusion $\mathcal{O}_L = \mathcal{O}_K[\alpha]$ says that the ring of integers of any finite local field extension is generated by a single element over the base ring. This is a strong statement: in general, rings of integers of number field extensions need not be monogenic, but the local setting guarantees it. The element $\alpha$ simultaneously lifts a primitive element of the residue field extension and, through the polynomial relation $f(\alpha) = \pi_L$, also generates the uniformizer of $L$. [/remark] The multiplicativity of $e$ and $f$ in towers is an immediate consequence. [quotetheorem:2346] [citeproof:2346] The practical force of multiplicativity is that it reduces questions about $e$ and $f$ for a large extension to the same questions for each step of a tower. In particular, to verify that $e_{M/K} = e_{L/K} \cdot e_{M/L}$, it suffices to check each intermediate step — one never needs to compute the full ramification index of $M/K$ directly. This decomposition principle will be essential when we study the inertia field: decomposing a general extension into an unramified layer (with $e = 1$) followed by a totally ramified layer (with $f = 1$) is exactly a tower to which multiplicativity applies. ## Unramified and Totally Ramified Extensions The fundamental identity $[L:K] = ef$ shows that an extension is determined, in a rough sense, by how it splits between the residue field and the "purely vertical" part. The two extreme cases — when all the extension is in the residue field, or none of it is — define the two main classes. [definition: Unramified Extension] Let $L/K$ be a finite extension of local fields. We say $L/K$ is **unramified** if $e_{L/K} = 1$, equivalently $f_{L/K} = [L:K]$. [/definition] [definition: Totally Ramified Extension] Let $L/K$ be a finite extension of local fields. We say $L/K$ is **totally ramified** if $f_{L/K} = 1$, equivalently $e_{L/K} = [L:K]$. [/definition] In an unramified extension, the prime of $K$ is still prime in $L$ — no ramification has occurred. The residue field grows by the full degree of the extension. In a totally ramified extension, the residue field does not change at all — the extension is "invisible" at the residue level — and the prime of $K$ becomes the $e$-th power of the new prime. Every extension of local fields can be decomposed as an unramified extension followed by a totally ramified extension; we shall see this after classifying each type. Before restricting to these extreme cases, it is worth seeing an extension that is neither. Consider $L = \mathbb{Q}_p(\zeta_{p^2-1}, \sqrt{p})$ over $K = \mathbb{Q}_p$. The extension $\mathbb{Q}_p(\zeta_{p^2-1})/\mathbb{Q}_p$ is unramified of degree $2$: the residue field of $\mathbb{Q}_p$ is $\mathbb{F}_p$, and $\mathbb{F}_{p^2}^\times$ is cyclic of order $p^2-1$, so all $(p^2-1)$-th roots of unity live in $\mathbb{F}_{p^2}$; Hensel's lemma lifts them uniquely to $\mathbb{Q}_p(\zeta_{p^2-1})$, which has $e = 1$ and $f = 2$. Adjoining $\sqrt{p}$ over this unramified extension is totally ramified of degree $2$: $\sqrt{p}$ satisfies $x^2 - p$, which is Eisenstein, so $e = 2$ and $f = 1$ for this top step. The composite $L/\mathbb{Q}_p$ therefore has degree $4$ with $e_{L/\mathbb{Q}_p} = 2$ and $f_{L/\mathbb{Q}_p} = 2$ — both invariants are strictly greater than $1$. This mixed behavior is the generic situation: the two extreme cases, unramified and totally ramified, are the building blocks into which every extension decomposes via its inertia field. ## Classification of Unramified Extensions Given a finite extension $\ell/k_K$ of the residue field, is there a unique unramified extension of $K$ realizing it? This classification problem has a remarkably clean answer: unramified extensions of $K$ correspond bijectively to finite extensions of the residue field $k_K$, and the correspondence is canonical. Since finite field extensions are completely understood — there is exactly one extension of $\mathbb{F}_q$ of each degree $n$, namely $\mathbb{F}_{q^n}$ — this gives a complete classification of all unramified extensions of $K$. [quotetheorem:2347] [citeproof:2347] The following lemma is the technical engine behind the classification. It says that maps between unramified extensions are completely controlled by maps between residue fields — there is no additional data to track. [quotetheorem:2348] [citeproof:2348] [citeproof:2348] [remark: The Frobenius Element] For $K = \mathbb{Q}_p$ and the unique unramified extension $L$ of degree $n$, the Galois group $\operatorname{Gal}(L/K) \cong \operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ is cyclic of order $n$, generated by the **Frobenius** $x \mapsto x^p$. The unique element of $\operatorname{Gal}(L/K)$ that acts as the Frobenius on the residue field is called the **arithmetic Frobenius** of $L/K$; it is the generator of the (cyclic) Galois group. The Frobenius plays a central role in class field theory. [/remark] The bijection on Hom-sets also gives an alternative description of the map $\varphi: L \to M$ from a residue map $\bar\varphi: k_L \to k_M$: using the uniformizer $\pi_K$ (which is also a uniformizer in $L$ since $L/K$ is unramified), one defines \begin{align*} \varphi\!\left(\sum_n [a_n]\pi_K^n\right) = \sum_n [\bar\varphi(a_n)]\pi_K^n, \end{align*} where $[-]$ denotes Teichmüller representatives. This makes the lifting completely explicit. Unramified extensions are stable under composita and subextensions. [quotetheorem:2349] [citeproof:2349] The theorem implies that among all subextensions of a given extension $L/K$, there is a canonical largest unramified one. [quotetheorem:2350] [citeproof:2350] The field $T$ is called the **inertia field** or **maximal unramified subextension** of $L/K$. The extension $L/T$ is then totally ramified (the residue fields of $T$ and $L$ coincide, so $f_{L/T} = 1$ and $e_{L/T} = [L:T]$). This decomposes any extension as an unramified base $T/K$ followed by a totally ramified top $L/T$.  ## Classification of Totally Ramified Extensions If an extension is totally ramified, what algebraic data determines it? In the unramified case the answer was clean: a finite field extension of the residue field. Here the answer is different in character — a totally ramified extension $L/K$ is generated by a uniformizer $\pi_L$, and the data that determines $L/K$ is the minimal polynomial of $\pi_L$ over $K$, which must be an Eisenstein polynomial. Totally ramified extensions are characterized by Eisenstein polynomials — a criterion most students have seen in algebraic number theory, but which has a particularly elegant proof in the local field setting using Newton polygons. [quotetheorem:2351] The proof uses Newton polygons and is a standard application of the theory developed in Chapter 2. The condition $\pi_K^2 \nmid a_0$ is essential: if $\pi_K^2 \mid a_0$, the Newton polygon argument breaks down and irreducibility can fail. For example, $f(x) = x^2 - p^2$ over $\mathbb{Q}_p$ satisfies $p \mid a_1 = 0$ and $p \mid a_0 = -p^2$, but $p^2 \mid a_0$, and $f$ is not irreducible — it factors as $(x-p)(x+p)$ over $\mathbb{Q}_p$ itself. Without the exact divisibility condition, the Newton polygon of $f$ has two segments of slope $-1$ instead of a single segment of slope $-1/n$, signaling that the roots split into two groups of equal valuation but permitting a factorization over $K$. [definition: Eisenstein Polynomial] A polynomial $f(x) \in \mathcal{O}_K[x]$ satisfying the conditions of the Eisenstein criterion — all non-leading coefficients divisible by $\pi_K$ and the constant term exactly divisible by $\pi_K$ (i.e., $v_K(a_0) = 1$) — is called an **Eisenstein polynomial** over $K$. [/definition] The characterization of totally ramified extensions via Eisenstein polynomials is one of the most useful tools in local field theory. [quotetheorem:2383] [citeproof:2383] [remark: The Integral Ring in Totally Ramified Extensions] In a totally ramified extension $L/K$, the ring of integers is particularly simple: since $k_L = k_K$, every element of $\mathcal{O}_L$ can be written as $\sum_{i \geq 0} a_i \pi_L^i$ where $a_i$ are lifted from $k_K \cong k_L$ — and the natural lift to $\mathcal{O}_K$ suffices. Thus $\mathcal{O}_L = \mathcal{O}_K[\pi_L]$, the simplest possible monogenic structure. This contrasts with the unramified case where the generator is a lift of the primitive element of the residue field extension. [/remark] [example: A Family of Totally Ramified Extensions of $\mathbb{Q}_p$] Let $K = \mathbb{Q}_p$ and fix $n \geq 1$. The polynomial $f(x) = x^n - p$ is Eisenstein at $p$: the constant term is $-p$ with $v_p(-p) = 1$, and all intermediate coefficients are $0$, which are divisible by $p$. So if $L = \mathbb{Q}_p(\pi_L)$ with $\pi_L^n = p$, then $L/\mathbb{Q}_p$ is totally ramified of degree $n$ with uniformizer $\pi_L$. We have $e_{L/\mathbb{Q}_p} = n$ and $k_L = k_{\mathbb{Q}_p} = \mathbb{F}_p$, confirming $f_{L/\mathbb{Q}_p} = 1$. The extension $\mathbb{Q}_p(\zeta_{p^n})/\mathbb{Q}_p$, where $\zeta_{p^n}$ is a primitive $p^n$-th root of unity, is another important family of totally ramified extensions. The minimal polynomial of $\zeta_{p^n}$ over $\mathbb{Q}_p$ is the $p^n$-th cyclotomic polynomial $\Phi_{p^n}(x) = (x^{p^n} - 1)/(x^{p^{n-1}} - 1)$, and the element $\pi = \zeta_{p^n} - 1$ is a uniformizer with Eisenstein minimal polynomial. These totally ramified extensions — the local analogues of cyclotomic extensions — play a fundamental role in local class field theory. [/example] ## The Valuation on an Extension How does the valuation on $K$ extend to $L$, and what is the relationship between the normalized valuations $v_K$ and $v_L$? The ramification index $e_{L/K}$ enters precisely as the scaling factor: the extension of $v_K$ to $L$ is $e_{L/K}^{-1} v_L$, so the two normalized valuations measure integrality at the same prime but at different resolutions — $v_L$ has finer granularity by a factor of $e_{L/K}$. [quotetheorem:2352] [citeproof:2352] This theorem is essential for working with totally ramified extensions: it gives a precise formula relating the two valuations and identifies the uniformizer as the element of smallest positive $w$-value. The description $e_{L/K}^{-1} = \min\{w(x) : x \in \mathfrak{m}_L\}$ makes transparent why totally ramified extensions have the full degree $n$ as their ramification index — the uniformizer $\pi_L$ of $L$ is an element of $w$-value $1/n$, and the valuation group $w(L^\times)$ is $\frac{1}{n}\mathbb{Z}$, which maps via $e_{L/K}^{-1} = 1/n$ back to $\frac{1}{n} \cdot \mathbb{Z}$ rather than $\mathbb{Z}$. Analysis on local fields exhibits surprising rigidity: power series behave differently, polynomials have multiple roots by default, and the Mahler basis replaces the monomial basis as the natural frame. Now we ask how Galois extensions of local fields fit together, turning to ramification theory to answer this question. # 5. Ramification Theory for Local Fields Chapter 5 classified extensions of local fields into unramified and totally ramified pieces, and showed that every finite extension decomposes as an unramified layer followed by a totally ramified one. But the totally ramified part carries far more internal structure than the single integer $e$ reveals. Within a totally ramified Galois extension $L/K$, the Galois group $\Gal(L/K)$ is filtered by the higher ramification groups $G_s(L/K)$, which measure how wildly the automorphisms move elements of $\mathcal{O}_L$. Studying these subgroups — how they sit inside one another, how their quotients look, and how they transform when passing to a subextension — is the subject of ramification theory proper. This chapter develops that theory, culminating in Herbrand's theorem, the two numbering schemes for ramification groups (lower and upper), and the statement of the Hasse-Arf theorem, which reveals a striking integrality property of the upper ramification jumps for abelian extensions. ## Filtrations on the Unit Group and the Galois Group Before studying the Galois group directly, it helps to have a parallel filtration on the unit group $U_K = \mathcal{O}_K^\times$ of a local field. The idea is to filter $U_K$ by how close a unit is to $1$: the higher up in the filtration, the closer to $1$. The Galois group will then be filtered in a compatible way, measuring how close an automorphism is to being the identity. [definition: Higher Unit Groups] Let $K$ be a local field with uniformizer $\pi_K$. The **higher unit groups** are \begin{align*} U_K^{(s)} = 1 + \pi_K^s \mathcal{O}_K \quad \text{for } s \geq 1, \end{align*} with the convention $U_K^{(0)} = U_K = \mathcal{O}_K^\times$. [/definition] These form a descending chain $U_K = U_K^{(0)} \supseteq U_K^{(1)} \supseteq U_K^{(2)} \supseteq \cdots$, where an element belongs to $U_K^{(s)}$ if and only if it is congruent to $1$ modulo $\mathfrak{m}_K^s$. The quotients of consecutive terms in this filtration are remarkably simple. [quotetheorem:2353] [citeproof:2353] The structure of these quotients will reappear when we compute the quotients of the higher ramification groups: the quotient $G_s(L/K)/G_{s+1}(L/K)$ injects into $U_L^{(s)}/U_L^{(s+1)} \cong k_L$, and this injection is the key tool for understanding the group-theoretic structure of $G_0(L/K)$. It is worth noting why the residue characteristic matters: the additive structure $k_L \cong \mathbb{F}_{p^f}$ is an elementary abelian $p$-group, which forces the quotients $G_s/G_{s+1}$ (for $s \geq 1$) to be $p$-groups. If the residue field had characteristic zero, the additive group $(k_L, +)$ would be torsion-free, and the quotients would be trivial — there would be no wild inertia at all. Wild ramification is therefore a purely positive-characteristic phenomenon, invisible over fields like $\mathbb{R}$ or $\mathbb{C}$ where the residue has characteristic zero. ## Higher Ramification Groups The inertia group $I(L/K) = G_0(L/K)$ tells us whether an extension is ramified, and the wild inertia $G_1(L/K)$ distinguishes tame from wild. But two wildly ramified extensions with isomorphic Galois groups and the same ramification index can behave very differently — for example, they can have different discriminants, different embedding problems, and different local Langlands parameters. The decomposition group and inertia group do not see this finer structure at all: a wildly ramified extension with a large inertia group could have its wild part concentrated in the first step ($G_1 \supsetneq G_2 = \{1\}$) or spread across many steps ($G_1 \supsetneq G_2 \supsetneq \cdots \supsetneq G_r \supsetneq G_{r+1} = \{1\}$), and the inertia group alone cannot distinguish these cases. To capture how "wild" the ramification truly is, we need a finer filtration than $G_{-1} \supseteq G_0 \supseteq G_1$: a whole descending chain of subgroups indexed by all integers $s \geq 0$, recording how many automorphisms fix $\mathcal{O}_L$ modulo successively higher powers of the maximal ideal. [definition: Higher Ramification Groups] Let $L/K$ be a finite Galois extension of local fields, and let $v_L$ be the normalized valuation of $L$. For $s \in \mathbb{R}_{\geq -1}$, the **$s$-th ramification group** (in the lower numbering) is \begin{align*} G_s(L/K) = \{\sigma \in \Gal(L/K) : v_L(\sigma(x) - x) \geq s + 1 \text{ for all } x \in \mathcal{O}_L\}. \end{align*} [/definition] The condition $v_L(\sigma(x) - x) \geq s + 1$ says that $\sigma$ moves every element of $\mathcal{O}_L$ by at most $\mathfrak{m}_L^{s+1}$. The larger $s$ is, the more tightly $\sigma$ is constrained to be close to the identity. We allow $s \in \mathbb{R}_{\geq -1}$ (rather than just integers) because fractional values will appear naturally when relating ramification groups across field extensions. Since $\sigma(x) - x \in \mathcal{O}_L$ for all $x \in \mathcal{O}_L$ (automorphisms preserve $\mathcal{O}_L$), the condition is vacuous when $s = -1$, so $G_{-1}(L/K) = \Gal(L/K)$. At $s = 0$, an element $\sigma \in G_0(L/K)$ satisfies $\sigma(x) \equiv x \pmod{\mathfrak{m}_L}$ for all $x \in \mathcal{O}_L$ — that is, $\sigma$ acts as the identity on the residue field $k_L$. [definition: Inertia Group] Let $L/K$ be a finite Galois extension of local fields. The **inertia group** of $L/K$ is \begin{align*} I(L/K) = G_0(L/K) = \ker\bigl(\Gal(L/K) \to \Gal(k_L/k_K)\bigr), \end{align*} the kernel of the natural reduction homomorphism. [/definition] The reduction map $\Gal(L/K) \to \Gal(k_L/k_K)$ is surjective: given the maximal unramified subextension $T/K$ of $L/K$, the Galois theory of $T/K$ gives a surjection $\Gal(L/K) \twoheadrightarrow \Gal(T/K) \xrightarrow{\sim} \Gal(k_L/k_K)$ (using $k_T = k_L$). So $G_0(L/K)$ is a normal subgroup of $\Gal(L/K)$ with quotient $\Gal(k_L/k_K)$, and the inertia group is trivial if and only if $L/K$ is unramified. The field fixed by $I(L/K)$ is the inertia field $T$, the maximal unramified subextension of $L/K$ encountered in Chapter 5. The Teichmüller lift interacts cleanly with the inertia group: if $\sigma \in I(L/K)$, then $\sigma([x]) = [x]$ for all $x \in k_L$ (where $[-]$ denotes the Teichmüller lift), and more generally $\sigma([x]) = [\bar\sigma(x)]$ where $\bar\sigma$ is the image of $\sigma$ in $\Gal(k_L/k_K)$. This follows from the uniqueness of the Teichmüller lift: the map $x \mapsto \sigma^{-1}([\bar\sigma(x)])$ is multiplicative, reduces to the identity on $k_L$, and so must equal the Teichmüller lift by its universal property. ## Structure of the Ramification Filtration The higher ramification groups form a filtration with a rich algebraic structure. The key theorem describes the quotients $G_s/G_{s+1}$ and shows they are abelian. [quotetheorem:2354] [citeproof:2354] This theorem has several important consequences. Since $G_s/G_{s+1}$ injects into $U_L^{(s)}/U_L^{(s+1)}$, combining with the quotient structure theorem gives: [quotetheorem:2355] [citeproof:2355] [definition: Wild Inertia Group and Tame Quotient] Let $L/K$ be a finite Galois extension of local fields with residue characteristic $p$. The **wild inertia group** is $G_1(L/K)$, and the **tame quotient** is $G_0(L/K)/G_1(L/K)$. [/definition] The names are explained by the following structural result. [quotetheorem:2356] [citeproof:2356] The terminology reflects a classical dichotomy in ramification theory. An extension is called **tamely ramified** if $G_1(L/K)$ is trivial (equivalently, the ramification index $e_{L/K}$ is not divisible by $p$) and **wildly ramified** if $G_1(L/K)$ is non-trivial (equivalently, $p \mid e_{L/K}$). Tame ramification is comparatively well-behaved; the wild case, controlled by $G_1$, is where genuine $p$-group phenomena enter. [example: Ramification Groups of a Cyclotomic Extension] Let $p$ be prime, $n \geq 1$, and let $K = \mathbb{Q}_p(\zeta_{p^n})$ where $\zeta_{p^n}$ is a primitive $p^n$-th root of unity. The extension $K/\mathbb{Q}_p$ is totally ramified of degree $p^{n-1}(p-1)$ with uniformizer $\pi = \zeta_{p^n} - 1$, so $\mathcal{O}_K = \mathbb{Z}_p[\zeta_{p^n}]$. The Galois group is $\Gal(K/\mathbb{Q}_p) \cong (\mathbb{Z}/p^n\mathbb{Z})^\times$, where $m \mapsto \sigma_m$ with $\sigma_m(\zeta_{p^n}) = \zeta_{p^n}^m$. To compute the ramification groups, we use the function $i_{K/\mathbb{Q}_p}(\sigma_m) = v_K(\sigma_m(\zeta_{p^n}) - \zeta_{p^n}) = v_K(\zeta_{p^n}^{m-1} - 1)$. If $m \equiv 1 \pmod{p^k}$ but $m \not\equiv 1 \pmod{p^{k+1}}$, then $\zeta_{p^n}^{m-1}$ is a primitive $p^{n-k}$-th root of unity. The minimal polynomial of a primitive $p^r$-th root of unity over $\mathbb{Q}_p$ evaluated near $1$ shows that $v_K(\zeta_{p^r} - 1) = p^{n-1}(p-1)/p^{r-1}(p-1) = p^{n-r}$. In particular: \begin{align*} v_K(\zeta_{p^n}^{m-1} - 1) = p^k. \end{align*} So $i_{K/\mathbb{Q}_p}(\sigma_m) = p^k$ where $k = v_p(m-1)$ (with $k = \infty$ when $m = 1$). The $s$-th ramification group is therefore \begin{align*} G_s(K/\mathbb{Q}_p) \cong \{m \in (\mathbb{Z}/p^n\mathbb{Z})^\times : m \equiv 1 \pmod{p^k}\}, \end{align*} for $p^k \geq s+1 > p^{k-1}$ (i.e., $k = \lceil \log_p(s+1) \rceil$). In other words, $G_s(K/\mathbb{Q}_p) \cong \Gal(K/\mathbb{Q}_p(\zeta_{p^k}))$. The lower numbering has jumps at $s = p^k - 1$ for $k = 1, 2, \ldots, n-1$. [/example] This example already suggests a problem with the lower numbering: the jump locations $p^k - 1$ grow exponentially, making the lower numbering unwieldy for comparing ramification across different extensions. The upper numbering will remedy this. ## The Function $i_{L/K}$ and Towers To understand how the ramification filtration behaves in towers of extensions, we introduce the function $i_{L/K}$ which captures the ramification data in a single number for each automorphism. [definition: The Function $i_{L/K}$] Let $L/K$ be a finite Galois extension of local fields. For $\sigma \in \Gal(L/K)$, define \begin{align*} i_{L/K}(\sigma) = \min_{x \in \mathcal{O}_L} v_L(\sigma(x) - x). \end{align*} By convention, $i_{L/K}(\mathrm{id}) = +\infty$. [/definition] With this notation, $G_s(L/K) = \{\sigma \in \Gal(L/K) : i_{L/K}(\sigma) \geq s+1\}$. The key computational tool is: [quotetheorem:2357] [citeproof:2357] This theorem is essential because it reduces $i_{L/K}(\sigma)$, which is defined as a minimum over all $x \in \mathcal{O}_L$, to a single computation involving a generator $\alpha$. The monogenic hypothesis $\mathcal{O}_L = \mathcal{O}_K[\alpha]$ is genuinely needed here: the proof works by writing every $x \in \mathcal{O}_L$ as a polynomial in $\alpha$ with coefficients in $\mathcal{O}_K$, then using the factorization $\sigma(\alpha)^i - \alpha^i = (\sigma(\alpha) - \alpha)(\sigma(\alpha)^{i-1} + \cdots + \alpha^{i-1})$ to bound $v_L(\sigma(x) - x)$ below by $v_L(\sigma(\alpha) - \alpha)$. Without a single generator, there is no such polynomial expression and the argument breaks down: if $\mathcal{O}_L$ requires at least two generators over $\mathcal{O}_K$, the minimum in the definition of $i_{L/K}(\sigma)$ may be achieved on a combination of generators that cannot be reduced to a single valuation calculation. The hypothesis is guaranteed by Chapter 5's result that the ring of integers of a local field extension is monogenic. The key application is the following formula relating $i_{L/K}$ to $i_{M/K}$ in a tower. [quotetheorem:2358] The factor $e_{M/L}^{-1}$ accounts for the difference between $v_L$ and $v_M$: since $v_M = e_{M/L} \cdot w$ where $w$ extends $v_L$, summing $i_{M/K}(\tau)$ counts in units of $v_M$, while $i_{L/K}$ counts in units of $v_L$. The real content is that the value $i_{L/K}(\sigma)$ equals the average of $i_{M/K}(\tau)$ over all lifts $\tau$ of $\sigma$ (weighted equally, after accounting for the normalization). This is a non-trivial divisibility statement: one must show that the product $a = \prod_{g \in H}(\tau g(\alpha) - \alpha)$ and $b = \sigma(\beta) - \beta$ satisfy $v_M(a) = v_M(b)$, where $\alpha$ generates $\mathcal{O}_M$ over $\mathcal{O}_L$ and $\beta$ generates $\mathcal{O}_L$ over $\mathcal{O}_K$. The proof establishes that $a \mid b$ and $b \mid a$ using polynomial identities. [citeproof:2358] ## Herbrand's Theorem and the $\eta$ Function The averaging formula for $i_{L/K}$ allows us to describe how ramification groups transform when we pass to subextensions. The result is Herbrand's theorem, which involves a certain function $\eta_{L/K}$ that serves as a "change of variables" between ramification indices. [definition: The Herbrand Function $\eta_{L/K}$] Let $L/K$ be a finite Galois extension of local fields with $G = \Gal(L/K)$ and $e_{L/K}$ the ramification index. For $s \in [-1, \infty)$, define \begin{align*} \eta_{L/K}(s) = e_{L/K}^{-1} \sum_{\sigma \in G} \min\bigl(i_{L/K}(\sigma),\, s+1\bigr) - 1. \end{align*} [/definition] The function $\eta_{L/K}$ can also be expressed as a piecewise-linear integral: [quotetheorem:2384] [citeproof:2384]  [quotetheorem:2359] [citeproof:2359] Herbrand's theorem is a fundamental compatibility between the ramification filtrations of a tower. It says that the subquotients of the ramification filtration of $M/K$ are "the same as" the ramification groups of $L/K$, after applying the change-of-variables $\eta_{M/L}$. ## Upper and Lower Numbering The Herbrand function $\eta_{L/K}$ is continuous, strictly increasing, satisfies $\eta_{L/K}(-1) = -1$ and $\eta_{L/K}(s) \to +\infty$ as $s \to +\infty$, so it is invertible. Its inverse $\psi_{L/K} = \eta_{L/K}^{-1}$ is used to reindex the ramification groups. [definition: Upper Numbering of Ramification Groups] Let $L/K$ be a finite Galois extension of local fields. The **upper numbering** of the ramification groups is \begin{align*} G^t(L/K) = G_{\psi_{L/K}(t)}(L/K), \quad t \in [-1, \infty), \end{align*} where $\psi_{L/K} = \eta_{L/K}^{-1}$. The original indexing $G_s(L/K)$ is called the **lower numbering**. [/definition] The upper numbering is compatible with quotients in towers in a much cleaner way than the lower numbering. [quotetheorem:2360] [citeproof:2360] This compatibility is the primary reason the upper numbering is the "right" one for global purposes. In the lower numbering, $G_s(M/K)H/H$ corresponds to $G_t(L/K)$ for $t = \eta_{M/L}(s) \neq s$ in general; in the upper numbering, $G^t(M/K)H/H = G^t(L/K)$ for the same $t$. The upper numbering passes cleanly to quotients without any change of index. [example: Upper Numbering in the Cyclotomic Case] Return to $K = \mathbb{Q}_p(\zeta_{p^n})$ over $\mathbb{Q}_p$. We computed the lower numbering: $G_s(K/\mathbb{Q}_p) \cong \Gal(K/\mathbb{Q}_p(\zeta_{p^k}))$ for $p^k - 1 \geq s > p^{k-1} - 1$. The jumps occur at $s = p^k - 1$ for $k = 1, \ldots, n-1$. Using the integral formula, the Herbrand function evaluates to $\eta_{K/\mathbb{Q}_p}(p^k - 1) = k$. Let us verify this. The integrand $1/(G_0 : G_x)$ is a step function: it equals $1/(p-1)$ for $0 < x < p-1$, equals $1/(p(p-1))$ for $p-1 < x < p^2-1$, and so on. Computing: \begin{align*} \eta_{K/\mathbb{Q}_p}(p^k - 1) &= (p-1)\cdot\frac{1}{p-1} + (p^2 - 1 - (p-1)) \cdot \frac{1}{p(p-1)} \\ &\quad + \cdots + (p^k - 1 - (p^{k-1} - 1)) \cdot \frac{1}{p^{k-1}(p-1)} \\ &= 1 + \frac{p(p-1)}{p(p-1)} + \cdots + \frac{p^{k-1}(p-1)}{p^{k-1}(p-1)} = k. \end{align*} Therefore, in the upper numbering: \begin{align*} G^k(K/\mathbb{Q}_p) = G_{\psi(k)}(K/\mathbb{Q}_p) = G_{p^k - 1}(K/\mathbb{Q}_p) \cong \Gal(K/\mathbb{Q}_p(\zeta_{p^k})). \end{align*} The upper numbering has jumps at the integers $k = 1, 2, \ldots, n-1$, and $G^k = \Gal(K/\mathbb{Q}_p(\zeta_{p^k}))$. This is far cleaner: the subfield $\mathbb{Q}_p(\zeta_{p^k})$ fixed by $G^k$ is directly indexed by the upper ramification number $k$. [/example] The cyclotomic example illustrates a general pattern: for abelian extensions, the upper numbering jumps only at integers. This is the content of the Hasse-Arf theorem. ## The Hasse-Arf Theorem The Hasse-Arf theorem is a deep result that constrains where the jumps in the upper ramification filtration can occur for abelian extensions. It has no analogue in the lower numbering. [quotetheorem:2361] In other words, all breaks in the upper ramification filtration of an abelian extension occur at non-negative integers. The proof of the Hasse-Arf theorem is difficult and uses the theory of local class field theory (to be developed in Chapters 7 and 8); we do not include it here. One approach is to use local class field theory to identify the upper ramification filtration of an abelian extension with certain filtrations on the multiplicative group $K^\times$, where integrality is visible from the structure of the unit groups $U_K^{(s)}$. [example: Non-abelian Extension Where the Upper Numbering Jumps at a Non-integer] Consider an $S_3$-extension of $\mathbb{Q}_2$. Let $F/\mathbb{Q}_2$ be the splitting field of $x^3 - 2$; this is a degree-6 extension with Galois group $S_3$, which is non-abelian. One can compute the lower ramification groups explicitly using the theory of extensions of local fields. A representative example: take the $S_3$-extension given by adjoining a root $\alpha$ of $x^3 - 2$ and a primitive cube root of unity $\omega$. The inertia group $G_0 \cong S_3$ (the extension is totally wildly ramified over $\mathbb{Q}_2$ since $6 = 2 \cdot 3$ and $2$ is the residue characteristic). Computing via the function $i_{F/\mathbb{Q}_2}(\sigma) = v_F(\sigma(\pi) - \pi)$ for a suitable uniformizer $\pi$, one finds that the lower numbering has jumps at $s = 1$ and $s = 3$, so $G_0 = G_1 \supsetneq G_2 = G_3 \supsetneq G_4 = \{1\}$, with $|G_1| = 6$, $|G_2| = |G_3| = 2$. Applying the Herbrand function: $\eta(s) = s$ for $0 \leq s \leq 1$, then $\eta(s) = 1 + (s-1)/3$ for $1 \leq s \leq 3$ (since $(G_0 : G_s) = 3$ on this range), giving $\eta(3) = 1 + 2/3 = 5/3$. The upper numbering therefore has a jump at $t = \eta(3) = 5/3$, which is not an integer. This is consistent: $S_3$ is not abelian, so the Hasse-Arf theorem does not apply, and non-integer jumps are exactly what one should expect. [/example] [remark: Why "Abelian" is Essential] The Hasse-Arf theorem fails without the abelian hypothesis. For non-abelian totally ramified extensions, the jumps of the upper numbering can occur at non-integer values, and the theorem provides no constraint. The cyclotomic example is abelian (since $(\mathbb{Z}/p^n\mathbb{Z})^\times$ is abelian), and accordingly all jumps occur at integers. The Hasse-Arf theorem is a refined version of the solvability of Galois groups of local field extensions, leveraging the abelian structure to extract integrality. [/remark] [remark: The Hasse-Arf Theorem and Class Field Theory] The Hasse-Arf theorem can be understood as a consequence of local class field theory (Chapter 7). In that theory, there is a reciprocity isomorphism $K^\times \xrightarrow{\sim} \Gal(L^{ab}/K)$ where $L^{ab}$ is the maximal abelian extension of $K$. Under this isomorphism, the filtration on $K^\times$ by the higher unit groups $U_K^{(s)}$ corresponds to the upper ramification filtration $G^t$. Since the filtration on $K^\times$ by $U_K^{(s)}$ has jumps at the integers $s = 0, 1, 2, \ldots$ (as seen from the structure theorem for quotients of higher unit groups), the upper ramification filtration of an abelian extension must also jump at integers. [/remark] ## The Different and Discriminant How do we detect the degree of ramification from the ideal structure of $\mathcal{O}_L$ alone, without computing automorphisms directly? The different and discriminant answer this question: they are two invariants that measure the ramification of an extension in an arithmetic way, tying the ramification groups to the ideal structure of $\mathcal{O}_L$. [definition: Complementary Module] Let $L/K$ be a finite separable extension of local fields. The **complementary module** (or **codifferent**) of $\mathcal{O}_L$ over $\mathcal{O}_K$ is \begin{align*} \mathcal{O}_L^\vee = \{x \in L : \operatorname{Tr}_{L/K}(x \mathcal{O}_L) \subseteq \mathcal{O}_K\}, \end{align*} where $\operatorname{Tr}_{L/K}: L \to K$ is the field trace. [/definition] Since $\mathcal{O}_L^\vee$ contains $\mathcal{O}_L$ (the trace of an element of $\mathcal{O}_L$ is an algebraic integer in $K$, hence in $\mathcal{O}_K$), and $\mathcal{O}_L^\vee$ is a fractional ideal of $\mathcal{O}_L$, it is of the form $\mathfrak{m}_L^{-d}$ for some $d \geq 0$. [definition: Different] The **different** $\mathfrak{D}_{L/K}$ of $L/K$ is the inverse of the complementary module: \begin{align*} \mathfrak{D}_{L/K} = (\mathcal{O}_L^\vee)^{-1} = \{x \in \mathcal{O}_L : x \mathcal{O}_L^\vee \subseteq \mathcal{O}_L\}. \end{align*} This is an ideal of $\mathcal{O}_L$. Its norm $N_{L/K}(\mathfrak{D}_{L/K})$ is the **discriminant** $\mathfrak{d}_{L/K}$, an ideal of $\mathcal{O}_K$. [/definition] The different has a beautiful description in terms of the minimal polynomial of a generator. If $\mathcal{O}_L = \mathcal{O}_K[\alpha]$ with minimal polynomial $f(x)$, then $\mathfrak{D}_{L/K} = f'(\alpha)\mathcal{O}_L$. This is the local analogue of the classical formula for the different in number fields. In terms of the ramification groups, the different is given by: [quotetheorem:2385] The formula $\sum_{s=0}^\infty (|G_s| - 1)$ counts, for each non-identity automorphism $\sigma$, the integer $i_{L/K}(\sigma) - 1$: since $\sigma \in G_s$ iff $i_{L/K}(\sigma) \geq s+1$, summing $|G_s| - 1$ over all $s \geq 0$ gives $\sum_{\sigma \neq 1} i_{L/K}(\sigma)$. The different thus encodes the total "weight" of all non-trivial automorphisms, measured by how much they move elements of $\mathcal{O}_L$. An unramified extension has all $G_s = \{1\}$ for $s \geq 1$ (and $G_0 = \{1\}$), so the sum is zero and $\mathfrak{D}_{L/K} = \mathcal{O}_L$, confirming that unramified extensions have trivial different. For a tamely ramified extension ($G_1 = \{1\}$), the sum is $|G_0| - 1 = e_{L/K} - 1$, giving $v_L(\mathfrak{D}_{L/K}) = e_{L/K} - 1$. Wild ramification (nontrivial $G_1$) contributes additional terms. [example: Different of a Totally Ramified Extension] Let $K = \mathbb{Q}_p$ and $L = \mathbb{Q}_p(\pi)$ with $\pi^n = p$, a totally ramified extension of degree $n$ with $(n, p) = 1$ (so the extension is tame). Then $G_0(L/K) = \Gal(L/K)$ (the extension is totally ramified), $|G_0| = n$, and $G_1 = \{1\}$ (since the extension is tamely ramified). The formula gives $v_L(\mathfrak{D}_{L/K}) = n - 1 = e_{L/K} - 1$. For comparison, using the generator $\pi$ with minimal polynomial $f(x) = x^n - p$, we compute $f'(x) = nx^{n-1}$, so $f'(\pi) = n\pi^{n-1}$. Thus $v_L(f'(\pi)) = v_L(n) + (n-1)v_L(\pi) = 0 + (n-1) = n-1$ (since $n$ is coprime to $p$, so $v_L(n) = 0$), confirming the formula. [/example] The unramified extensions of a local field are completely determined by degree: they correspond to extensions of the residue field via a canonical isomorphism. Totally ramified extensions, by contrast, are far richer in structure; their study consumes the remainder of ramification theory. # 6. Further Ramification Theory The preceding chapters developed all the necessary local machinery: valuations, ramification theory, the structure of the residue field, norm maps, and the cohomological tools for studying extensions. Local class field theory is the reward: it classifies all abelian extensions of a local field $K$ in terms of the multiplicative group $K^\times$. The central object is the **local Artin map** (also called the **local reciprocity map**), a canonical isomorphism from $K^\times$ to the Weil group of the maximal abelian extension of $K$. The existence theorem then shows that every open finite-index subgroup of $K^\times$ arises as a norm group of a unique finite abelian extension. The chapter closes with the local Kronecker–Weber theorem, which makes the abstract theory entirely explicit for $\mathbb{Q}_p$. ## Infinite Galois Theory and Profinite Groups For finite extensions, the Galois group is a finite group with clear structure. But local class field theory requires the maximal abelian extension — an infinite extension. What topology must we put on $\operatorname{Gal}(M/K)$ to recover a meaningful Galois correspondence? Throughout this section, $K$ is any field. [definition: Separable and Normal Extensions] Let $L/K$ be an algebraic extension of fields. We say $L/K$ is **separable** if, for every $\alpha \in L$, the minimal polynomial $f_\alpha \in K[x]$ is separable (has no repeated roots in any algebraic closure). We say $L/K$ is **normal** if $f_\alpha$ splits completely in $L$ for every $\alpha \in L$. [/definition] [definition: Galois Extension] An algebraic extension $L/K$ is **Galois** if it is both normal and separable. When $L/K$ is Galois, we write \begin{align*} \operatorname{Gal}(L/K) = \operatorname{Aut}_K(L). \end{align*} [/definition] These definitions agree with the usual finite-case definitions. The distinction begins when we try to formulate the Galois correspondence. In the finite case, subgroups of $\operatorname{Gal}(L/K)$ biject with intermediate fields. In the infinite case, $\operatorname{Gal}(L/K)$ has far too many subgroups — most of them do not correspond to any subfield. To see the failure concretely, consider $G = \operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ with the discrete topology. Choose a basis for $\mathbb{Q}_p^{ur}$ over $\mathbb{Q}_p$ as a $\mathbb{Q}_p$-vector space, and construct a $\mathbb{Q}_p$-linear automorphism of $\mathbb{Q}_p^{ur}$ that sends some basis element to a transcendental (over the prime field); extend it arbitrarily. The stabilizer of a single element in such a construction defines a subgroup $H \leq G$ that is not closed — its fixed field is not even a field extension of $\mathbb{Q}_p$ in any sensible sense. The naive "subgroups $\leftrightarrow$ subfields" bijection simply breaks down: there are vastly more subgroups of $G$ than there are algebraic intermediate fields. The fix is to equip $\operatorname{Gal}(L/K)$ with a topology and restrict attention to closed subgroups. [definition: Krull Topology] Let $M/K$ be a Galois extension. The **Krull topology** on $\operatorname{Gal}(M/K)$ is defined by taking the cosets $\{\sigma \operatorname{Gal}(M/L) : L/K \text{ finite}\}$ as a basis of open sets. Explicitly, $U \subseteq \operatorname{Gal}(M/K)$ is open if for every $\sigma \in U$, there exists a finite subextension $L/K$ of $M/K$ such that $\sigma \operatorname{Gal}(M/L) \subseteq U$. [/definition] [remark: Open Subgroups Are Closed] Any open subgroup of a topological group is automatically closed. When $M/K$ is finite, the Krull topology is discrete, since one may take the finite subextension to be $M$ itself. [/remark] The Galois group of an infinite extension is a **profinite group**: a topological group that is compact and Hausdorff, and in which every open neighbourhood of the identity contains an open normal subgroup. [quotetheorem:2386] [citeproof:2386] Compactness is not a vacuous topological nicety — it is what makes the infinite Galois correspondence work. In a compact group, every closed subgroup is an intersection of open subgroups, and the correspondence between closed subgroups and fixed fields is a genuine bijection. Without compactness, the topology would fail to capture enough algebraic information about the extension. The presence of arbitrarily small open normal subgroups (the second part of the theorem) reflects the fact that any automorphism is "determined by its action on finitely many elements," a key feature of algebraic extensions. To describe $\operatorname{Gal}(M/K)$ algebraically, we use the inverse limit construction, which expresses the infinite Galois group as the limit of all its finite quotients. [definition: Directed System] A partially ordered set $(I, \leq)$ is a **directed system** if for all $i, j \in I$, there exists $k \in I$ with $i \leq k$ and $j \leq k$. [/definition] Any total order is a directed system; so is $\mathbb{N}$ with divisibility. The key example here is the set of finite Galois subextensions of an infinite Galois extension, ordered by inclusion. [definition: Inverse System and Inverse Limit] Let $I$ be a directed system. An **inverse system** of topological groups indexed by $I$ consists of topological groups $G_i$ for each $i \in I$ and continuous homomorphisms $f_{ij} : G_j \to G_i$ for all $i \leq j$, satisfying $f_{ii} = \operatorname{id}_{G_i}$ and $f_{ik} = f_{ij} \circ f_{jk}$ whenever $i \leq j \leq k$. The **inverse limit** of the system $(G_i, f_{ij})$ is \begin{align*} \varprojlim_{i \in I} G_i = \left\{(g_i) \in \prod_{i \in I} G_i : f_{ij}(g_j) = g_i \text{ for all } i \leq j\right\}, \end{align*} with coordinatewise multiplication and the subspace topology from the product topology on $\prod_{i \in I} G_i$. This makes $\varprojlim_{i \in I} G_i$ a topological group. [/definition] [quotetheorem:2362] This theorem makes precise the slogan "an automorphism of the infinite extension is the same as a compatible system of automorphisms of all finite subextensions." The inverse limit description is not just elegant: it gives a concrete handle on an otherwise unwieldy object. For $K = \mathbb{Q}_p$, the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ is an inverse limit of finite groups $\operatorname{Gal}(L/\mathbb{Q}_p)$ ranging over all finite Galois extensions $L$. Each factor is a finite group we can in principle compute; the inverse limit packages them all into one profinite group. The Krull topology is precisely the topology that makes this isomorphism a homeomorphism, not merely a group isomorphism. [quotetheorem:2387] [citeproof:2387] The theorem is a satisfying vindication of the topological approach: the Galois correspondence does extend to infinite extensions, but only when framed in terms of closed subgroups. The subtlety — and this cannot be overstated — is that the bijection genuinely fails for all subgroups: an abstract group like $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$ has uncountably many subgroups, but only countably many closed ones (indexed by the divisors of a supernatural number). The finite-extension condition translating to openness is another striking feature: open subgroups of a profinite group have finite index, which corresponds exactly to the field $M^H$ being a finite extension of $K$. ## Unramified Extensions and the Frobenius Among all extensions of $K$, which have the simplest ramification? The unramified ones — but to classify them all at once, we need the infinite unramified extension $K^{ur}$ and its canonical generator, the Frobenius. Let $K$ be a local field throughout this section. [definition: Infinite Unramified and Totally Ramified Extensions] An algebraic extension $M/K$ is **unramified** if every finite subextension $L/K$ of $M/K$ is unramified. It is **totally ramified** if every finite subextension is totally ramified. [/definition] Note that an infinite unramified extension need not be a local field, since its residue field may be infinite. [quotetheorem:2363] [citeproof:2363] This result is remarkable for how completely it identifies the unramified part of the Galois theory with the Galois theory of residue fields. In particular, for the maximal unramified extension $K^{ur}$, we get $\operatorname{Gal}(K^{ur}/K) \cong \operatorname{Gal}(\overline{k}/k)$, where $k = k_K$ is the residue field. For local fields with finite residue field $\mathbb{F}_q$, this gives $\operatorname{Gal}(K^{ur}/K) \cong \hat{\mathbb{Z}}$, the profinite completion of $\mathbb{Z}$, topologically generated by the Frobenius. The entire unramified part of the absolute Galois group is thus determined by a single element. Since composites of unramified extensions are unramified, every local field $K$ has a **maximal unramified extension** $K^{ur}$, which is the union of all unramified extensions inside a fixed algebraic closure. More generally, any algebraic extension $M/K$ has a maximal unramified subextension $T = T_{M/K}$. For a finite unramified extension $L/K$, the residue field extension $k_L/k_K$ is a finite extension of finite fields. By standard field theory, $\operatorname{Gal}(k_L/k_K)$ is cyclic, generated by the $q$-power Frobenius $x \mapsto x^q$ where $q = |k_K|$. [definition: Arithmetic Frobenius] Let $L/K$ be a finite unramified extension of local fields. The **arithmetic Frobenius** $\operatorname{Frob}_{L/K} \in \operatorname{Gal}(L/K)$ is the unique element whose image under the isomorphism $\operatorname{Gal}(L/K) \cong \operatorname{Gal}(k_L/k_K)$ is the map $x \mapsto x^{|k_K|}$. [/definition] [remark: Geometric Frobenius] There is also a geometric Frobenius, which is the inverse $\operatorname{Frob}_{L/K}^{-1}$. We use only the arithmetic Frobenius in this course. [/remark] The Frobenius is compatible with towers: if $M/L/K$ is a tower of finite unramified extensions, then $\operatorname{Frob}_{M/K}|_L = \operatorname{Frob}_{L/K}$. This compatibility means the system $(\operatorname{Frob}_{L/K})_{L/K \text{ finite unramified}}$ defines an element \begin{align*} \operatorname{Frob}_{M/K} \in \varprojlim_{L/K} \operatorname{Gal}(L/K) \cong \operatorname{Gal}(M/K) \end{align*} for any unramified extension $M/K$. In particular, $K^{ur}$ has a distinguished element $\operatorname{Frob}_K = \operatorname{Frob}_{K^{ur}/K} \in \operatorname{Gal}(K^{ur}/K)$, which is the unique lift of $x \mapsto x^{|k_K|}$ to $K^{ur}$. For a finite unramified extension, $\operatorname{Frob}_{L/K}$ generates the entire Galois group. For infinite unramified extensions, the Frobenius no longer generates the whole Galois group (the closure of the subgroup it generates is the whole group, but the Frobenius alone only generates an infinite cyclic dense subgroup). The Weil group is designed to retain exactly the information carried by integer powers of the Frobenius. [definition: Weil Group] Let $K$ be a local field and $M/K$ Galois. Let $T = T_{M/K}$ be the maximal unramified subextension of $M/K$. The **Weil group** of $M/K$ is \begin{align*} W(M/K) = \{\sigma \in \operatorname{Gal}(M/K) : \sigma|_T = \operatorname{Frob}_{T/K}^n \text{ for some } n \in \mathbb{Z}\}. \end{align*} We give $W(M/K)$ the topology in which $U$ is open if and only if there exists a finite extension $L/T$ such that $\sigma \operatorname{Gal}(L/T) \subseteq U$. [/definition] The topology on $W(M/K)$ is chosen so that $\operatorname{Gal}(M/T)$ sits inside $W(M/K)$ as an open normal subgroup with the Krull topology, while the quotient $W(M/K)/\operatorname{Gal}(M/T) \cong \mathbb{Z}$ carries the discrete topology. This makes all maps in the diagram \begin{align*} \operatorname{Gal}(M/T) \hookrightarrow W(M/K) \twoheadrightarrow \operatorname{Frob}_{T/K}^{\mathbb{Z}} \end{align*} continuous. In particular, if $M/K$ is unramified, then $W(M/K) = \operatorname{Frob}_{T/K}^{\mathbb{Z}} \cong \mathbb{Z}$. [quotetheorem:2364] [citeproof:2364] Density is what allows the Weil group to serve as a replacement for the full Galois group in the formulation of class field theory. The Galois group $\operatorname{Gal}(M/K)$ is a profinite group and is in particular uncountable (as a set) when $M$ is an infinite extension; the Weil group $W(M/K)$ is only a dense subgroup, but it carries strictly more structure — its quotient by $\operatorname{Gal}(M/T)$ is $\mathbb{Z}$ rather than $\hat{\mathbb{Z}}$, keeping track of the integer "Frobenius power" rather than just the profinite completion. This distinction is the key technical reason why the local Artin map lands in $W(K^{ab}/K)$ rather than in $\operatorname{Gal}(K^{ab}/K)$: the target must have room for an isomorphism from $K^\times$, and $K^\times$ contains a copy of $\mathbb{Z}$ (the valuations) that must map to integer Frobenius powers. ## The Local Artin Map The central question of local class field theory: given a local field $K$, can we classify all its finite abelian extensions using only the internal structure of $K^\times$? The answer is yes — through a canonical map $\operatorname{Art}_K: K^\times \to W(K^{ab}/K)$. Fix a local field $K$ and an algebraic closure $\bar{K}$; all algebraic extensions are subextensions of $\bar{K}/K$. Write $K^{sep}$ for the separable closure inside $\bar{K}$. Why can we not classify abelian extensions by brute force? There are infinitely many, with no obvious parametrization. For $K = \mathbb{Q}_p$, there is one unramified extension of each degree $n \geq 1$, a continuum of totally ramified extensions (parametrized by Eisenstein polynomials), and composites thereof. Without a structural principle, listing them is hopeless: how would one decide whether two Eisenstein polynomials of the same degree define the same or different extensions, or whether a given totally ramified extension is abelian? Without the norm group structure, one would have no way to distinguish the three quadratic extensions of $\mathbb{Q}_2$ from one another — they are all degree-$2$ extensions, all totally ramified, with Galois group $\mathbb{Z}/2\mathbb{Z}$, yet they are genuinely distinct. The Artin map resolves this by replacing the field-theoretic question with a group-theoretic one: abelian extensions of $K$ correspond to open finite-index subgroups of $K^\times$, and the latter are completely classified by the structure $K^\times \cong \mathbb{Z} \times \mathcal{O}_K^\times$. [definition: Abelian Extension] A Galois extension $L/K$ is **abelian** if $\operatorname{Gal}(L/K)$ is abelian. [/definition] The composite of two abelian extensions is abelian: if $L/K$ and $M/K$ are both abelian Galois, the restriction map $\operatorname{Gal}(LM/K) \hookrightarrow \operatorname{Gal}(L/K) \times \operatorname{Gal}(M/K)$ shows that $\operatorname{Gal}(LM/K)$ is abelian. Consequently there exists a **maximal abelian extension** $K^{ab}$: the composite of all abelian extensions inside $\bar{K}$. Note that $K^{ur} \subseteq K^{ab}$, since every unramified extension is abelian. We write $\operatorname{Frob}_K = \operatorname{Frob}_{K^{ur}/K} \in \operatorname{Gal}(K^{ur}/K)$. [quotetheorem:2365] This is the central theorem of local class field theory; its proof is substantial and uses cohomological machinery (Brauer groups, local Tate duality, and the computation of $H^2(K^\times)$). The theorem as stated is proved in courses on local cohomology; here we take it as given and draw out its consequences. Let us unpack what the theorem says. Property (1) pins down the behaviour of $\operatorname{Art}_K$ on the unramified part: every uniformizer maps to the Frobenius restricted to the unramified extension. Remarkably, this holds for **any** uniformizer — changing the uniformizer by a unit does not change the Frobenius. Property (2) says that norms from abelian extensions act as the identity, which is the reciprocity law in its most distilled form. The functoriality relation makes the Artin map compatible with field extensions. For a finite abelian extension $L/K$, the last isomorphism specialises to give a surjection $K^\times \twoheadrightarrow \operatorname{Gal}(L/K)$ with kernel exactly $N_{L/K}(L^\times)$. Consequently: [quotetheorem:2388] [citeproof:2388] The index bound is a key structural fact: it says the norm group can never have index larger than the degree, and it achieves its maximum index precisely for abelian extensions. For a non-abelian extension $L/K$, the norm group $N(L/K)$ is strictly larger than $N((L \cap K^{ab})/K)$, meaning some norms from the non-abelian part "collapse" — they are already norms from a smaller abelian extension. This is why non-abelian extensions are invisible to the Artin map: their norm groups coincide with those of their maximal abelian subextensions, and the Artin map can only see the abelian part. [example: Unramified Extensions and Frobenius] Let $L/K$ be the unique unramified extension of degree $n$. Then $\operatorname{Gal}(L/K) \cong \mathbb{Z}/n\mathbb{Z}$, generated by $\operatorname{Frob}_{L/K}$. By property (1) of the Artin map, $\operatorname{Art}_K(\pi_K)$ restricts to $\operatorname{Frob}_{L/K}$. The norm group $N(L/K) \subseteq K^\times$ consists of those elements $x \in K^\times$ with $v_K(x) \equiv 0 \pmod{n}$, together with all units that are norms of units in $\mathcal{O}_L^\times$. In the simplest case $K = \mathbb{Q}_p$ and $n = 2$: $N(L/\mathbb{Q}_p)$ consists of $p$-adic numbers of even valuation whose unit part is a norm from the unique unramified quadratic extension of $\mathbb{Z}_p$. [/example] ## The Existence Theorem and Norm Groups The Local Artin Reciprocity theorem tells us that each finite abelian extension $L/K$ produces an open finite-index subgroup $N(L/K) \leq K^\times$. The existence theorem establishes the converse: every open finite-index subgroup arises this way. To state it precisely, we define a partial order on both sides. For subgroups of $K^\times$, inclusion gives a partial order. For finite abelian extensions, containment of fields gives a partial order. The norms reverse the order: if $L \subseteq M$, then $N(M/K) \subseteq N(L/K)$. [quotetheorem:2366] The existence theorem completes the picture: together with Local Artin Reciprocity, it establishes a perfect duality between the lattice of open finite-index subgroups of $K^\times$ and the lattice of finite abelian extensions of $K$, with the lattice operations translating exactly as the norm-group identities show. The composite $LM$ corresponds to the intersection of norm groups — the larger the field, the smaller the norm group — while the intersection $L \cap M$ corresponds to the product of norm groups. This is a local shadow of the global class field theory correspondence between ideal class groups and abelian extensions of number fields. The theorem has no simple elementary proof; it relies on the full strength of local cohomology, specifically the computation of the Brauer group $\operatorname{Br}(K)$ and the local invariant map. One key input is that $H^2(G_K, K^{sep,\times}) \cong \mathbb{Q}/\mathbb{Z}$, which pins down the cohomological dimension of the absolute Galois group and forces the existence of the required extensions. The upshot — that every open finite-index subgroup of $K^\times$ is a norm group — is genuinely surprising: there is no a priori reason why an algebraically defined subgroup (norms from an extension) should recover every open subgroup of the topological group $K^\times$. A partial result toward the existence theorem follows directly from Local Artin Reciprocity: [quotetheorem:2389] [citeproof:2389] [remark: Why "Class Field Theory"?] The field $(K^{ab})^{\operatorname{Art}_K(H)}$ corresponding to an open subgroup $H \leq K^\times$ is called the **class field** of $H$. The name comes from the global theory, where the analogous groups on the multiplicative side are "class groups" of number fields. In the local case, $K^\times$ itself plays the role of the class group. [/remark] ## The Local Kronecker–Weber Theorem The abstract theory of the Artin map becomes entirely concrete for $K = \mathbb{Q}_p$. The local Kronecker–Weber theorem asserts that every abelian extension of $\mathbb{Q}_p$ is contained in a cyclotomic extension — the local analogue of the classical Kronecker–Weber theorem for $\mathbb{Q}$. To set the stage, recall that for a primitive $m$-th root of unity $\zeta_m$, the extension $\mathbb{Q}_p(\zeta_m)/\mathbb{Q}_p$ is abelian because $\operatorname{Gal}(\mathbb{Q}_p(\zeta_m)/\mathbb{Q}_p)$ injects into $(\mathbb{Z}/m\mathbb{Z})^\times$ via the action on $\zeta_m$. We also need to understand the norm groups of cyclotomic extensions. Write $m = p^a \cdot r$ with $(r, p) = 1$. The extension $\mathbb{Q}_p(\zeta_{p^a})/\mathbb{Q}_p$ is totally ramified of degree $p^{a-1}(p-1)$, while $\mathbb{Q}_p(\zeta_r)/\mathbb{Q}_p$ is unramified of degree equal to the order of $p$ modulo $r$. [quotetheorem:2367] [citeproof:2367] [example: Abelian Extensions of $\mathbb{Q}_2$] For $p = 2$, the structure is slightly different because $\mathbb{Z}_2^\times \cong \{\pm 1\} \times (1 + 4\mathbb{Z}_2)$. The extension $\mathbb{Q}_2(\sqrt{-1}) = \mathbb{Q}_2(\zeta_4)$ is totally ramified of degree $2$, with norm group $\{x \in \mathbb{Q}_2^\times : x \equiv 1 \pmod{4}\} \cup \{x : v_2(x) \text{ even}, x/2^{v_2(x)} \equiv 1 \pmod 4\}$. The three quadratic extensions of $\mathbb{Q}_2$ are $\mathbb{Q}_2(\sqrt{-1})$, $\mathbb{Q}_2(\sqrt{2})$, and $\mathbb{Q}_2(\sqrt{-2})$, corresponding via the Artin map to the three index-$2$ subgroups of $\mathbb{Q}_2^\times / (\mathbb{Q}_2^\times)^2 \cong (\mathbb{Z}/2\mathbb{Z})^3$. [/example] [explanation: The Artin Map on Units] The Local Artin Reciprocity theorem shows that $\operatorname{Art}_K$ restricts to a map on units $\mathcal{O}_K^\times \to W(K^{ab}/K)$. The image of $\mathcal{O}_K^\times$ is exactly $\operatorname{Gal}(K^{ab}/K^{ur})$, the inertia group of the maximal abelian extension. This is a reflection of property (1): a uniformizer $\pi_K$ maps to the Frobenius, which acts as the identity on any totally ramified extension. Conversely, elements of $\mathcal{O}_K^\times$ have valuation $0$, so they do not "see" the unramified part at all — they act trivially on $K^{ur}$ and their image lies entirely within the inertia subgroup. More precisely, the Artin map gives a commutative diagram: \begin{align*} 1 \to \mathcal{O}_K^\times \to K^\times \xrightarrow{v_K} \mathbb{Z} \to 0 \\ \downarrow \operatorname{Art}_K \quad\quad\quad \downarrow \operatorname{Art}_K \quad\quad \downarrow n \mapsto \operatorname{Frob}_K^n \\ 1 \to \operatorname{Gal}(K^{ab}/K^{ur}) \to W(K^{ab}/K) \to \mathbb{Z} \to 0 \end{align*} with exact rows, where the bottom row uses the discrete topology on $\mathbb{Z}$ and the Weil group construction. This exhibits the Artin map as precisely interleaving the ramified and unramified parts of $K^\times$ with the corresponding pieces of the Weil group. [/explanation] Ramification and inertia have been classified, the different has been computed, and we have seen how extensions interact with absolute value and residue field. All these ingredients now assemble into local class field theory, which describes the abelian extensions of a local field through its local Artin map. # 7. Local Class Field Theory This chapter completes the course by delivering on the promise of local class field theory: we construct the maximal abelian extension $K^{\mathrm{ab}}$ and the local Artin map $\operatorname{Art}_K$ explicitly. The main tool is Lubin–Tate theory, which associates to each uniformizer $\pi \in \mathcal{O}_K$ a family of totally ramified abelian extensions of $K$. Together with the unramified extensions, these generate all of $K^{\mathrm{ab}}$. The approach is more hands-on than the Galois-cohomological one, and gives a precise formula for the Artin map in terms of the action on torsion points of formal groups. We close by computing the higher ramification groups of the Lubin–Tate extensions, which provides the explicit reciprocity law linking the filtration of $K^\times$ to the ramification filtration of $\operatorname{Gal}(K^{\mathrm{ab}}/K)$. ## A Motivating Computation for $\mathbb{Q}_p$ Before developing the general theory, it is instructive to work out what the Artin map must look like for $K = \mathbb{Q}_p$ with uniformizer $p$. This computation will guide the construction of the general case. Recall from Chapter 7 that for any local field $K$, there is a topological decomposition \begin{align*} K^\times \cong \langle \pi_K \rangle \times \mathcal{O}_K^\times, \end{align*} and the finite abelian extensions of $K$ correspond bijectively to open finite-index subgroups of $K^\times$ via their norm groups. The key structural subgroups are $\langle \pi_K^m \rangle \times U_K^{(n)}$. [quotetheorem:2487] [citeproof:2487] [remark: Unramified Extensions via Norms] An immediate consequence: $L/K$ is unramified if and only if $N_{L/K}(\mathcal{O}_L^\times) = \mathcal{O}_K^\times$. This confirms the earlier characterization—unramified extensions are completely determined by their residue field extensions and do not disturb the unit group. [/remark] The norm group of any open subgroup of finite index in $K^\times$ contains some $\langle \pi_K^m \rangle \times U_K^{(n)}$. Since $N(LM/K) = N(L/K) \cap N(M/K)$, it suffices to construct the abelian extensions corresponding to the two special families: - $\langle \pi_K^m \rangle \times \mathcal{O}_K^\times$ — these are the unramified extensions, already handled by the residue field theory; - $\langle \pi_K \rangle \times U_K^{(n)}$ — these are totally ramified extensions that depend on the choice of $\pi_K$. [quotetheorem:2486] [quotetheorem:2485] [citeproof:2485] For $K = \mathbb{Q}_p$, the totally ramified abelian extensions with norm group $\langle p \rangle \times U^{(n)}$ are the cyclotomic extensions $\mathbb{Q}_p(\zeta_{p^n})$. [example: Cyclotomic Extensions and the Artin Map for $\mathbb{Q}_p$] Set $L_n = \mathbb{Q}_p(\zeta_{p^n})$, where $\zeta_{p^n}$ is a primitive $p^n$-th root of unity. By a direct computation (using the $p$-adic cyclotomic polynomial), the norm group is \begin{align*} N(\mathbb{Q}_p(\zeta_{p^n})/\mathbb{Q}_p) = \langle p \rangle \times (1 + p^n \mathbb{Z}_p) = \langle p \rangle \times U_{\mathbb{Q}_p}^{(n)}, \end{align*} confirming this is a totally ramified extension. Setting $\mathbb{Q}_p(\zeta_{p^\infty}) = \bigcup_n \mathbb{Q}_p(\zeta_{p^n})$, the Galois group is computed via the inverse limit: \begin{align*} \operatorname{Gal}(\mathbb{Q}_p(\zeta_{p^\infty})/\mathbb{Q}_p) \cong \varprojlim_n \operatorname{Gal}(\mathbb{Q}_p(\zeta_{p^n})/\mathbb{Q}_p) = \varprojlim_n (\mathbb{Z}/p^n\mathbb{Z})^\times = \mathbb{Z}_p^\times. \end{align*} The transition maps in the inverse limit are the standard reduction maps $(\mathbb{Z}/p^n\mathbb{Z})^\times \to (\mathbb{Z}/p^{n-1}\mathbb{Z})^\times$, consistent with the restriction maps on Galois groups. To read off the Artin map, let $m \in \mathbb{Z}_p^\times$ with $p$-adic expansion $m = a_0 + a_1 p + \cdots$. Under the chain of isomorphisms, $m$ acts on $\mathbb{Q}_p(\zeta_{p^n})$ as \begin{align*} \zeta_{p^n} \mapsto \zeta_{p^n}^m. \end{align*} However, the Artin map sends $m$ to the inverse of this automorphism. Writing $\sigma_m$ for the automorphism $\zeta_{p^n} \mapsto \zeta_{p^n}^m$, the full Artin map is determined by the commutative diagram: \begin{align*} \operatorname{Art}_{\mathbb{Q}_p}: \mathbb{Q}_p^\times \cong \langle p \rangle \times \mathbb{Z}_p^\times &\longrightarrow \operatorname{Frob}_{\mathbb{Q}_p}^\mathbb{Z} \times \operatorname{Gal}(\mathbb{Q}_p(\zeta_{p^\infty})/\mathbb{Q}_p)\\ (p^n, m) &\longmapsto (\operatorname{Frob}_{\mathbb{Q}_p}^n, \sigma_{m^{-1}}). \end{align*} The inverse in $\sigma_{m^{-1}}$ is a normalization choice ensuring compatibility with the uniformizer condition $\operatorname{Art}(\pi)|_{K^{\mathrm{ur}}} = \operatorname{Frob}_K$. [/example] [quotetheorem:2368] This is the special case $K = \mathbb{Q}_p$ of the generalized local Kronecker–Weber theorem we prove later (see the generalization below). The first line says every abelian extension of $\mathbb{Q}_p$ is obtained by adjoining some root of unity, and the second identifies the unramified part precisely as the roots of unity of order coprime to $p$. [remark: Normalization and the Geometric Frobenius] There is an alternative normalization of the Artin map in which a uniformizer is sent to the geometric Frobenius $\operatorname{Frob}^{-1}$. Under this convention, $\operatorname{Art}_{\mathbb{Q}_p}(m)|_{\mathbb{Q}_p(\zeta_{p^\infty})} = \sigma_m$ (without the inverse). Both normalizations appear in the literature; the one used here (uniformizer maps to arithmetic Frobenius) is more common in the explicit Lubin–Tate approach. [/remark] ## Formal Groups To construct the totally ramified extensions for a general local field $K$, we need the language of formal groups. A formal group is a power series that encodes a group law — or rather, the infinitesimal neighborhood of the identity in a group — without needing an underlying geometric space. The key insight is that for a complete ring, convergence of formal power series on the maximal ideal is automatic, so the formal group law becomes a genuine group structure on $\mathfrak{m}_K$. [definition: Formal Group] A (one-dimensional, commutative) **formal group** over a ring $R$ is a power series $F(X, Y) \in R[[X, Y]]$ satisfying: 1. $F(X, Y) \equiv X + Y \pmod{(X^2, XY, Y^2)}$ (i.e., the tangent map at the identity is the addition map); 2. $F(X, Y) = F(Y, X)$ (commutativity); 3. $F(X, F(Y, Z)) = F(F(X, Y), Z)$ (associativity). [/definition] The condition modulo degree-two terms says the formal group looks like ordinary addition at first order. The higher terms encode the deviation from being additive. [example: Formal Additive and Multiplicative Groups] The simplest examples are: - The **formal additive group**: $\hat{\mathbb{G}}_a(X, Y) = X + Y$. This is the formal group whose law is literally addition. - The **formal multiplicative group**: $\hat{\mathbb{G}}_m(X, Y) = X + Y + XY = (1+X)(1+Y) - 1$. If $K$ is a complete valued field, there is a bijection $\mathfrak{m}_K \leftrightarrow 1 + \mathfrak{m}_K$ sending $x \mapsto 1 + x$. Under this bijection, the multiplicative group structure of $1 + \mathfrak{m}_K$ transports to the formal group law $\hat{\mathbb{G}}_m$ on $\mathfrak{m}_K$. [/example] The name "formal group" is justified because identity elements and inverses are automatic: [quotetheorem:2390] The proof, which proceeds by successive approximation, is set as an exercise; the key point is that uniqueness forces the coefficients of $i(X)$ to be determined one by one. When $K$ is a complete valued field, a formal group $F$ over $\mathcal{O}_K$ produces a genuine group structure on $\mathfrak{m}_K$: since both inputs have absolute value strictly less than $1$, the power series $F(x,y)$ converges in the complete field $K(x,y)$ (a finite extension of $K$) and the result lies in $\mathfrak{m}_{K(x,y)} \subseteq \bar{\mathfrak{m}}$. The next step is to equip a formal group with a ring action. [definition: Homomorphism of Formal Groups] Let $R$ be a ring and $F, G$ be formal groups over $R$. A **homomorphism** $f: F \to G$ is a power series $f \in R[[X]]$ with $f(X) \equiv 0 \pmod{X}$ such that \begin{align*} f(F(X, Y)) = G(f(X), f(Y)). \end{align*} The endomorphisms $\mathrm{End}_R(F)$ form a ring, where addition is \begin{align*} (f +_F g)(X) = F(f(X), g(X)), \end{align*} and multiplication is composition. [/definition] [definition: Formal Module] A **formal $R$-module** is a formal group $F$ over $R$ together with a ring homomorphism $R \to \mathrm{End}_R(F)$, written $a \mapsto [a]_F$, satisfying \begin{align*} [a]_F(X) = aX + O(X^2). \end{align*} [/definition] The condition $[a]_F(X) \equiv aX \pmod{X^2}$ says the action of $a \in R$ on $\mathfrak{m}_K$ (via $F$) has derivative $a$ at $0$, as expected for a module action. ## The Lubin–Tate Construction What kind of formal group gives us the right torsion points to generate totally ramified abelian extensions? We need a formal group that "knows about" the uniformizer $\pi$ — specifically, one whose $\pi$-multiplication map reduces to the Frobenius mod $\pi$. Without this link, the torsion points would generate extensions with no special relationship to $\pi$, and the Artin map would have no reason to send $\pi$ to the Frobenius. The Lubin–Tate modules are exactly the formal groups with this property. We now specialize to $R = \mathcal{O}_K$ for a local field $K$ with residue field of order $q = |\kappa_K|$ and uniformizer $\pi$. [definition: Lubin–Tate Module] A **Lubin–Tate module** over $\mathcal{O}_K$ with respect to $\pi$ is a formal $\mathcal{O}_K$-module $F$ such that \begin{align*} [\pi]_F(X) \equiv X^q \pmod{\pi}. \end{align*} [/definition] The condition $[\pi]_F(X) \equiv X^q \pmod{\pi}$ says the action of $\pi$ on the mod-$\pi$ reduction looks like the Frobenius. This is the key link between the uniformizer and the arithmetic Frobenius, which we already saw in the $\mathbb{Q}_p$ example: the formal multiplicative group $\hat{\mathbb{G}}_m$ is a Lubin–Tate module for $p$. [example: $\hat{\mathbb{G}}_m$ as a Lubin–Tate $\mathbb{Z}_p$-module] The formal group $\hat{\mathbb{G}}_m$ with the action \begin{align*} [a]_{\hat{\mathbb{G}}_m}(X) = (1+X)^a - 1 = \sum_{n=1}^\infty \binom{a}{n} X^n, \quad a \in \mathbb{Z}_p, \end{align*} is a Lubin–Tate $\mathbb{Z}_p$-module with respect to $p$. We verify: $[a]_{\hat{\mathbb{G}}_m}(X) \equiv aX \pmod{X^2}$ holds since $\binom{a}{1} = a$. The Lubin–Tate condition is $(1+X)^p - 1 \equiv X^p \pmod{p}$, which follows from the binomial theorem since all binomial coefficients $\binom{p}{k}$ for $1 \leq k \leq p-1$ are divisible by $p$. The ring homomorphism property follows from $(1+X)^{ab} = ((1+X)^a)^b$ and $(1+X)^{a+b} = (1+X)^a (1+X)^b$. [/example] To construct and classify all Lubin–Tate modules, we introduce the following: [definition: Lubin–Tate Series] A **Lubin–Tate series** for $\pi$ is a power series $e(X) \in \mathcal{O}_K[[X]]$ satisfying \begin{align*} e(X) \equiv \pi X \pmod{X^2}, \qquad e(X) \equiv X^q \pmod{\pi}. \end{align*} The set of all Lubin–Tate series for $\pi$ is denoted $\mathcal{E}_\pi$. [/definition] By definition, if $F$ is a Lubin–Tate module for $\pi$, then $[\pi]_F \in \mathcal{E}_\pi$. The most concrete example is: [definition: Lubin–Tate Polynomial] A **Lubin–Tate polynomial** for $\pi$ is a polynomial of the form \begin{align*} e(X) = u X^q + \pi(a_{q-1} X^{q-1} + \cdots + a_2 X^2) + \pi X, \end{align*} where $u \in U_K^{(1)} = 1 + \mathfrak{m}_K$ (a principal unit) and $a_2, \ldots, a_{q-1} \in \mathcal{O}_K$. In particular, Lubin–Tate polynomials are Lubin–Tate series. [/definition] [example: Standard Lubin–Tate Polynomials] The simplest Lubin–Tate polynomial is $e(X) = X^q + \pi X$. For $K = \mathbb{Q}_p$ and $\pi = p$, the polynomial $(1+X)^p - 1 = X^p + pX^{p-1} + \cdots + pX$ is also a Lubin–Tate polynomial. [/example] The classification of all Lubin–Tate modules rests on a key existence and uniqueness lemma, which we state without proof: [quotetheorem:2391] The proof proceeds by successive approximation: the conditions uniquely force each homogeneous component of $F$ one degree at a time. The two conditions together — linearity at first order, and equivariance under the Lubin–Tate series — are strong enough to pin down $F$ completely. Before proceeding, it is worth understanding why the formal additive group $\hat{\mathbb{G}}_a$ does not work here. The additive group satisfies $[\pi]_{\hat{\mathbb{G}}_a}(X) = \pi X$, which reduces mod $\pi$ to $0$, not to $X^q$. So $\hat{\mathbb{G}}_a$ is not a Lubin–Tate module for $\pi$. More concretely, the torsion points of $\hat{\mathbb{G}}_a$ are the roots of $\pi^n X = 0$, i.e., $X = 0$. There are no non-trivial $\pi^n$-torsion points! This is because the additive group over a field of characteristic $0$ has no torsion — it is isomorphic to $(\bar{K}, +)$, which is torsion-free. The Lubin–Tate condition $[\pi]_F(X) \equiv X^q \pmod{\pi}$ is precisely what forces the torsion points to be plentiful and arithmetically interesting. From this lemma, two important corollaries follow by specializing: [quotetheorem:2392] The non-triviality of this theorem lies in the two-condition uniqueness: the first condition (linear at first order) fixes the "tangent map" of $F_e$, while the second (equivariance under $e$) ties $F_e$ to the arithmetic of $\pi$. Neither condition alone determines $F_e$ — together they pin down every coefficient inductively. The intertwining maps $[a]_{e_1, e_2}$ are equally remarkable: they show that Lubin–Tate modules for the same $\pi$ but different series are not just isomorphic in some abstract sense, but explicitly related by a canonical power series in $\mathcal{O}_K[[X]]$. [quotetheorem:2393] [citeproof:2393] The isomorphism here is not merely an abstract existence result. The isomorphism $[a]_{e_1, e_2}$ between any two Lubin–Tate modules for the same $\pi$ depends only on the unit $a \in \mathcal{O}_K^\times$, and in particular does not depend on any choice made over an extension of $K$. This means the family of field extensions $K(F(n))$ is truly an invariant of $\pi$, not of the particular series $e$ we used to construct $F$. ## Division Points and the Lubin–Tate Extensions How do we extract field extensions from a formal group? The key is to adjoin the torsion points — the elements killed by some power of $[\pi]_F$. This is exactly the formal-group analogue of adjoining roots of unity to generate cyclotomic extensions. The Lubin–Tate structure ensures these points generate totally ramified extensions, with the Galois group acting via the ring of endomorphisms. We now use Lubin–Tate modules to construct explicit abelian extensions of $K$. Fix an algebraic closure $\bar{K}$ and let $\bar{\mathfrak{m}} = \mathfrak{m}_{\bar{K}}$. [quotetheorem:2369] [citeproof:2369] [definition: Division Points] Let $F$ be a Lubin–Tate $\mathcal{O}_K$-module for $\pi$ and $n \geq 1$. The group of **$\pi^n$-division points** is \begin{align*} F(n) = \{x \in \bar{\mathfrak{m}}_F : [\pi^n]_F(x) = 0\} = \ker([\pi^n]_F). \end{align*} This is both a group under $+_F$ and an $\mathcal{O}_K$-submodule of $\bar{\mathfrak{m}}_F$. [/definition] The terminology mirrors the analogous notion for elliptic curves: just as the $n$-torsion of an elliptic curve consists of points killed by the multiplication-by-$n$ map, the division points are the torsion elements of the formal module. [example: Division Points of $\hat{\mathbb{G}}_m$] For $F = \hat{\mathbb{G}}_m$ over $\mathbb{Z}_p$ with $\pi = p$, the action is $[p^n]_F(x) = (1+x)^{p^n} - 1$. So: \begin{align*} \hat{\mathbb{G}}_m(n) = \{x \in \bar{\mathfrak{m}} : (1+x)^{p^n} = 1\} = \{\zeta_{p^n}^i - 1 : i = 0, 1, \ldots, p^n - 1\}, \end{align*} where $\zeta_{p^n} \in \bar{\mathbb{Q}}_p$ is a primitive $p^n$-th root of unity. Thus $\hat{\mathbb{G}}_m(n)$ generates the extension $\mathbb{Q}_p(\zeta_{p^n})$, which is the $n$-th cyclotomic field. [/example] To show that $F(n)$ has the expected size, we first establish that the iterated Lubin–Tate series has no repeated roots: [quotetheorem:2370] [citeproof:2370] The importance of separability for Galois theory is immediate: if $f_n$ had repeated roots, the splitting field $L_n = K(F(n))$ could fail to be Galois over $K$, and the identification $\operatorname{Gal}(L_n/K) \cong \mathrm{Aut}_{\mathcal{O}_K}(F(n))$ would break down. The separability is also what forces the bound $|F(n)| = q^n$: a polynomial of degree $q^n$ with all simple roots contributes exactly $q^n$ elements. [quotetheorem:2371] [citeproof:2371] The freeness of rank $1$ is crucial: it means $F(n)$ is a cyclic $\mathcal{O}_K/\pi^n$-module, so a single generator $\lambda_n$ determines everything. In particular, $K(\lambda_n) = K(F(n)) = L_n$ — the entire $\pi^n$-torsion is already defined over the field generated by a single generator. This is the key input to the Galois group calculation: the Galois action on all of $F(n)$ is determined by where it sends $\lambda_n$, and that image must be another generator, i.e., $[u]_F(\lambda_n)$ for some unit $u \in U_K/U_K^{(n)}$. [quotetheorem:2372] The first isomorphism says every endomorphism is given by multiplication by an element of $\mathcal{O}_K/\pi^n$; the second says the invertible ones correspond to the units modulo $U_K^{(n)}$. ## The Main Theorem on Lubin–Tate Extensions What is the Galois group of the extension generated by $\pi^n$-division points? The answer connects the formal module structure to the Galois action in the most direct possible way: every Galois automorphism acts on the torsion module $F(n)$ as multiplication by some unit $u \in U_K$, and this correspondence is an isomorphism. This is the formal-group analogue of the classical fact that cyclotomic Galois groups are isomorphic to groups of roots of unity. Given a Lubin–Tate module $F$ for $\pi$, define the field of $\pi^n$-division points: \begin{align*} L_n = L_{n,\pi} = K(F(n)). \end{align*} From the inclusions $F(n) \subseteq F(n+1)$, we get a tower $L_n \subseteq L_{n+1}$. Note that $L_n$ depends only on $\pi$, not on the choice of $F$: if $G$ is another Lubin–Tate module with isomorphism $f: F \to G$ (defined over $K$), then $G(n) = f(F(n)) \subseteq K(F(n))$, so $K(G(n)) \subseteq K(F(n))$, and by symmetry we have equality. [quotetheorem:2394] [citeproof:2394] The explicit formula $\sigma(\lambda) = [u]_F(\lambda)$ is the heart of the matter. It says the Galois action is not some mysterious permutation of roots, but multiplication by a unit in the formal module structure. This is directly analogous to how elements of $(\mathbb{Z}/p^n)^\times$ act on $p^n$-th roots of unity by raising them to powers — the Lubin–Tate theory makes this analogy exact for any local field. Notice also the role of the tower: the subgroup $\operatorname{Gal}(L_m/L_n) \cong U_K^{(n)}/U_K^{(m)}$ consists of automorphisms that fix $L_n$, i.e., those acting by units $u \equiv 1 \pmod{\pi^n}$ — exactly the units whose effect on $F(n)$ is trivial. [example: Cyclotomic Extensions Revisited] For $K = \mathbb{Q}_p$, $\pi = p$, we have $\hat{\mathbb{G}}_m(n) = \{\zeta_{p^n}^i - 1\}$. The theorem gives \begin{align*} \operatorname{Gal}(\mathbb{Q}_p(\zeta_{p^n})/\mathbb{Q}_p) \cong (\mathbb{Z}/p^n)^\times, \end{align*} where $a \in (\mathbb{Z}/p^n)^\times$ acts by $\sigma_a(\zeta_{p^n}^i - 1) = [a]_{\hat{\mathbb{G}}_m}(\zeta_{p^n}^i - 1) = (1 + \zeta_{p^n}^i - 1)^a - 1 = \zeta_{p^n}^{ai} - 1$. This agrees with the classical cyclotomic theory. [/example] ## The Maximal Abelian Extension and the Artin Map Do the Lubin–Tate extensions, together with the unramified extensions, account for ALL abelian extensions of $K$? This is the central question, and the answer is yes — but proving it requires showing that no totally ramified abelian extension can "escape" the tower $L_1 \subseteq L_2 \subseteq \cdots$. The key constraint comes from the Hasse–Arf theorem (Chapter 6): in any abelian extension, the jumps in the upper ramification filtration are integers. The Lubin–Tate extensions already achieve all integer jumps, leaving no room for anything else. Setting $L_\infty = \bigcup_{n=1}^\infty L_n$, the extension $L_\infty/K$ is Galois with \begin{align*} \operatorname{Gal}(L_\infty/K) \cong \varprojlim_n \operatorname{Gal}(L_n/K) \cong \varprojlim_n U_K/U_K^{(n)} \cong U_K. \end{align*} The map $\sigma \mapsto (\sigma|_{L_n})_n$ gives this isomorphism explicitly. This will be the inverse of the Artin map restricted to $U_K$. The naive approach of generating totally ramified extensions by taking $\pi^n$-th roots fails for general $K$. If we simply adjoin a root of $X^{p^n} - \pi$ to $K$, the resulting extension is indeed totally ramified — but it is not abelian in general. The Galois closure of $K(\pi^{1/p^n})$ typically has Galois group that is a non-abelian extension of a cyclic group, because the $p^n$-th roots of unity $\zeta_{p^n}$ are not in $K$ (for a general local field). For $K = \mathbb{Q}_p$ this accident does not occur: $\mathbb{Q}_p(\zeta_{p^n}, p^{1/p^n})$ is indeed abelian over $\mathbb{Q}_p(\zeta_{p^n})$, but the extension $\mathbb{Q}_p(\zeta_{p^n})/\mathbb{Q}_p$ itself is needed. For $K = \mathbb{Q}_p(\zeta_{p^n})$ (for instance), the cyclotomic roots are already present and $p^{1/p^n}$ does not generate an abelian extension of $K$. The Lubin–Tate construction avoids this pitfall by using the formal group torsion, which automatically generates abelian extensions because the Galois action is encoded in the endomorphism ring of the formal module. To show that $L_\infty$ accounts for all totally ramified abelian extensions, we invoke: [quotetheorem:2373] The proof of this result uses the Hasse–Arf theorem, which states that in an abelian extension the jumps in the upper ramification filtration occur only at integer values. One shows that $L_\infty$ already realizes all possible integer jumps, so no additional abelian extension can exist. Alternatively, one can first construct the Artin map independently (e.g., via Galois cohomology) and then use the norm group calculation to conclude. With $K^{\mathrm{ab}} = K^{\mathrm{ur}} L_\infty$ established, it remains to make the Artin map fully explicit. For this we need to compute the norm groups of the Lubin–Tate extensions — knowing the norm group is equivalent to knowing the abelian extension, by the fundamental theorem of local class field theory. The norm group of $L_n$ is identified as follows: [quotetheorem:2395] The proof uses Coleman operators (power series representations of the norm map) or appeals to the Artin map once it is constructed. The key inputs are $N_{L_n/K}(-\lambda_n) = \pi$ (from the Eisenstein polynomial) and the fact that the Galois group acts on $F(n)$ via units in $U_K/U_K^{(n)}$, forcing $U_K^{(n)}$ to be in the kernel of the reciprocity map restricted to $L_n$. The norm group $\langle \pi \rangle \times U_K^{(n)}$ has index $q^{n-1}(q-1) = [L_n : K]$ in $K^\times$, which is exactly right. As $n \to \infty$, these norm groups shrink to $\langle \pi \rangle \times \{1\}$, corresponding to $L_\infty$ being the kernel of the projection $K^\times \to \langle \pi \rangle \cong \mathbb{Z}$. Together, the norm groups $\langle \pi \rangle \times U_K^{(n)}$ for $n \geq 1$ form a cofinal family that pins down $L_\infty$ as the totally ramified part of $K^{\mathrm{ab}}$. With all this in place, the Artin map is: [quotetheorem:2396] The inverse $u^{-1}$ in the formula is necessary to ensure independence of the choice of uniformizer: replacing $\pi$ by $\pi' = u_0 \pi$ (for $u_0 \in U_K$) changes the isomorphism $K^\times \cong \langle \pi \rangle \times U_K$, and the inverse compensates for this change. The Artin map we have constructed is completely explicit: given any $x = \pi^m u \in K^\times$ (with $u \in U_K$), the automorphism $\operatorname{Art}_K(x)$ acts on the unramified part via $\operatorname{Frob}_K^m$ and on the Lubin–Tate part via $[u^{-1}]_F$. There is nothing opaque or non-constructive here. This stands in contrast to the Galois-cohomological approach, where the Artin map exists by abstract duality and its explicit form requires additional work to extract. The price we pay is that the Lubin–Tate approach is more computational and requires the formal group machinery developed above. ## Higher Ramification Groups of Lubin–Tate Extensions What does the ramification filtration look like for Lubin–Tate extensions? The explicit nature of the construction allows us to compute it. Because we know every Galois automorphism acts as $\sigma_u: \lambda \mapsto [u]_F(\lambda)$ for some unit $u$, we can directly measure how much $\sigma_u$ moves a uniformizer $\lambda$ — the difference $\sigma_u(\lambda) - \lambda$ — and read off the ramification break from the valuation of that difference. We conclude by computing the ramification filtration of $\operatorname{Gal}(L_n/K)$, which makes the explicit reciprocity law completely transparent. [quotetheorem:2397] [citeproof:2397] The sizes of the lower ramification groups tell us the "depth" of ramification at each level. The group $G_s(L_n/K)$ has order $q^{n-k}$ for $q^{k-1}-1 < s \leq q^k - 1$, shrinking by a factor of $q$ each time we move up to the next break. This explains why the breaks occur at $s = q^k - 1$ for $k = 1, \ldots, n-1$: these are the values where the conductor of the norm character jumps, reflecting the precise depth at which a unit $u = 1 + \varepsilon \pi^k$ first acts nontrivially on $F(n)$ but trivially on $F(n-k)$. [quotetheorem:2374] [citeproof:2374] The contrast between lower and upper numbering is stark. In the lower numbering, the ramification groups jump at $s = q^k - 1$, which depends on $q$ and grows exponentially with $k$. In the upper numbering, the jumps occur at $t = 1, 2, \ldots, n-1$ — one unit step at a time, regardless of $q$. The Herbrand change-of-variable exactly "normalizes away" the size of the residue field. [remark: Elegance of Upper Numbering] The upper numbering restores a clean picture: the ramification filtration of $\operatorname{Gal}(L_n/K) \cong U_K/U_K^{(n)}$ jumps at integers $0, 1, 2, \ldots, n-1$, with $G^t = \operatorname{Gal}(L_n/L_{\lceil t \rceil})$ for $-1 \leq t \leq n-1$. Under the Artin map, $\operatorname{Art}_K^{-1}(G^t(L_n/K)) = U_K^{(\lceil t\rceil)}/U_K^{(n)}$. By passage to the limit, $\operatorname{Art}_K^{-1}(G^t(K^{\mathrm{ab}}/K)) = U_K^{(\lceil t \rceil)}$. [/remark] ## The Explicit Reciprocity Law What is the precise relationship between the ramification filtration and the unit group filtration under the Artin map? The answer, once we have the ramification computation in hand, is as clean as one could hope: the $t$-th ramification group of $K^{\mathrm{ab}}/K$ maps back under $\operatorname{Art}_K^{-1}$ to exactly the $\lceil t \rceil$-th unit group $U_K^{(\lceil t \rceil)}$. The upper numbering turns what was an irregular filtration in the lower numbering into one that lines up perfectly with the filtration of $K^\times$. The ramification computation culminates in the following complete description of the abelian extensions of $K$: [quotetheorem:2375] [citeproof:2375] This is the culmination of everything built in this chapter. The Explicit Reciprocity Law says the ramification filtration of $\operatorname{Gal}(K^{\mathrm{ab}}/K)$ is the Artin image of the unit group filtration $\{U_K^{(n)}\}$ — no information is lost or gained. For the abstract local class field theory, this is a non-trivial theorem requiring significant machinery. The Lubin–Tate approach makes it a computation: we built the extensions explicitly, computed the ramification directly, and read off the correspondence. [quotetheorem:2376] [citeproof:2376] In concrete terms: if $M = L_n$ is the $n$-th Lubin–Tate extension, then $N(L_n/K) = \langle \pi \rangle \times U_K^{(n)}$, and the formula gives $G^t(L_n/K) = \operatorname{Art}_K(U_K^{(\lceil t\rceil)}/U_K^{(n)}) = \operatorname{Gal}(L_n/L_{\lceil t\rceil})$, recovering the earlier calculation. The point is that this now holds for any finite abelian $M/K$, not just the Lubin–Tate extensions themselves. This theorem is the culmination of the course. It says the ramification filtration of $\operatorname{Gal}(M/K)$ is completely determined by the norm group $N(M/K)$ and the filtration $\{U_K^{(n)}\}$ of $K^\times$: the $t$-th ramification subgroup is the image of the $\lceil t\rceil$-th unit group modulo the norm group. The Lubin–Tate construction has made the abstract existence theorem of local class field theory entirely explicit. ## References These notes are based on lectures by H. C. Johansson (Cambridge, Michaelmas 2016). Standard references for the subject include J.-P. Serre, *Local Fields*; J. W. S. Cassels, *Local Fields*; and J. Neukirch, *Algebraic Number Theory*.

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