Number fields form the natural generalization of the integers within algebraic number theory, extending the familiar ring of integers to finite extensions of the rationals. This course develops the fundamental theory of algebraic integers and their arithmetic properties, revealing how many classical results about the ordinary integers — such as unique factorization — can be understood and refined through the lens of algebraic structures. The course begins by establishing what number fields are and introducing the ring of integers within them, then systematically develops the tools needed to understand their multiplicative structure and the arithmetic of their ideals.

The central theme is the interplay between algebraic and arithmetic properties: we study how ideals factor into primes, how norms and traces relate elements to rational integers, and how the fundamental group-theoretic quantities — the class number and unit group — constrain the overall arithmetic of the field. The chapters build progressively from concrete objects (number fields and their basic invariants) to structural results, culminating in finiteness theorems that assert the class group is finite and the unit group is finitely generated. Along the way, the Minkowski bound provides a geometric bridge between ideals and lattice points, offering a tool to compute class groups explicitly for specific fields.

The final chapter on $L$-functions and Dirichlet series connects algebraic number theory to analytic methods, showing how the analytic properties of certain generating functions encode deep arithmetic information about the field. Throughout, the course emphasizes both the theory — understanding why these structures exist and what they measure — and explicit computation, developing techniques to work with concrete examples and compute invariants that govern the arithmetic of a given number field.

# 1. Number fields

This chapter introduces the central objects of the course: number fields and their rings of integers. Starting from concrete arithmetic questions about which integers can be expressed as sums of two squares, we arrive at the abstract framework of algebraic integers and integral ring extensions. The key algebraic engine is a Cayley–Hamilton argument that drives the proof that $\mathcal{O}_L$ is a ring — a fact that would be trivial if the extension were a vector space over a field, but requires genuine work for module extensions over $\mathbb{Z}$.

## From Arithmetic Questions to Number Fields

The motivating question is deceptively simple: which integers can be written as $x^2 + y^2$ for $x, y \in \mathbb{Z}$? And if $a$ and $b$ are both of this form, must $ab$ be as well?

The elegant answer comes from working inside $\mathbb{Z}[i]$. The integers expressible as sums of two squares are precisely the norms of elements of $\mathbb{Z}[i]$: for $\alpha = x + iy$,

\begin{align*} N(x + iy) &= |x + iy|^2 = x^2 + y^2. \end{align*}

The multiplicativity of the norm then gives the result immediately: if $a = N(z)$ and $b = N(w)$, then $ab = N(zw)$.

A second example: the solutions to the Diophantine equation $x^2 + 2 = y^3$ are found by working in $\mathbb{Z}[\sqrt{-2}]$. Both examples illustrate the same technique — embed an integer arithmetic problem into a larger ring $\mathcal{O}_L$ sitting inside a field extension $L$ of $\mathbb{Q}$, leverage the ring structure there, then descend back to conclusions about $\mathbb{Z}$.

[definition: Field Extension] A **field extension** is an inclusion of fields $K \subseteq L$. We write this as $L/K$. [/definition]

[definition: Degree of a Field Extension] Let $K \subseteq L$ be a field extension. Then $L$ is a vector space over $K$, and the **degree** of the extension is \begin{align*} [L : K] &= \dim_K(L). \end{align*} If $[L : K]$ is finite, the extension is called **finite**. [/definition]

The degree records the size of the extension; it is the most fundamental numerical invariant we attach to $L/K$. We now single out the class of extensions the course studies.

[definition: Number Field] A **number field** is a finite field extension of $\mathbb{Q}$. [/definition]

The examples $\mathbb{Q}(i)$ and $\mathbb{Q}(\sqrt{-2})$ are both number fields of degree 2. Degree alone does not determine the field: $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{-1})$ are both quadratic extensions but are structurally different.

## Algebraic Integers

A field is the simplest kind of ring: its only ideals are $\{0\}$ and the field itself. If we want interesting algebraic structure inside a number field, we must look for a distinguished subring. For $\mathbb{Q}$, that subring is $\mathbb{Z}$. What is the correct analogue in a general number field $L$?

One naive attempt: take the $\mathbb{Z}$-span of a $\mathbb{Q}$-basis of $L$. This gives a $\mathbb{Z}$-module inside $L$, but it depends on the choice of basis and is generally not a ring. Another attempt: call $\alpha \in L$ an "integer" if $\alpha$ satisfies some polynomial with integer coefficients. But non-monic polynomials allow fractions: $\frac{1}{2}$ satisfies $2x - 1 = 0$, and we certainly don't want $\frac{1}{2} \in \mathcal{O}_\mathbb{Q}$. This suggests the right criterion: restrict to elements satisfying a monic polynomial with integer coefficients. The monic requirement forces the leading term to be $1$, cutting off the denominators that admitted $\frac{1}{2}$. Equivalently, monic polynomials in $\mathbb{Z}[x]$ are precisely those for which the leading coefficient does not introduce denominators when one uses the relation to express high powers of $\alpha$ in terms of lower powers.

[definition: Algebraic Integer] Let $L$ be a number field. An **algebraic integer** in $L$ is an element $\alpha \in L$ for which there exists a monic polynomial $f \in \mathbb{Z}[x]$ with $f(\alpha) = 0$. The set of all algebraic integers in $L$ is denoted $\mathcal{O}_L$. [/definition]