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]