Number Fields and Rings of Integers - Main Theorem

The fundamental structure theorem characterizes rings of integers as finitely generated modules over $\mathbb{Z}$ with optimal properties.

TheoremCharacterization of Rings of Integers

Let $K$ be a number field of degree $n$ . Then:

$\mathcal{O}_K$ is a free $\mathbb{Z}$ -module of rank $n$
$\mathcal{O}_K$ is the integral closure of $\mathbb{Z}$ in $K$
$\mathcal{O}_K$ is a Dedekind domain
The field of fractions of $\mathcal{O}_K$ is $K$

Moreover, there exists an integral basis $\{\omega_1, \ldots, \omega_n\}$ such that every element of $\mathcal{O}_K$ can be uniquely written as $\sum_{i=1}^n a_i \omega_i$ with $a_i \in \mathbb{Z}$ .

The freeness of $\mathcal{O}_K$ as a $\mathbb{Z}$ -module is crucial: it means $\mathcal{O}_K \cong \mathbb{Z}^n$ as abelian groups. This structure allows us to study arithmetic in $K$ using lattice theory and geometry of numbers.

TheoremTrace Pairing

The trace pairing $\langle \alpha, \beta \rangle = \text{Tr}_{K/\mathbb{Q}}(\alpha\beta)$ defines a non-degenerate symmetric bilinear form on $K$ as a $\mathbb{Q}$ -vector space.

For $\alpha, \beta \in \mathcal{O}_K$ , we have $\langle \alpha, \beta \rangle \in \mathbb{Z}$ . The discriminant $\Delta_K$ equals the determinant of the Gram matrix of the trace pairing with respect to any integral basis.

ExampleComputing Integral Bases

For $K = \mathbb{Q}(\sqrt{d})$ with $d$ square-free:

If $d \equiv 2, 3 \pmod{4}$ : integral basis is $\{1, \sqrt{d}\}$
If $d \equiv 1 \pmod{4}$ : integral basis is $\left\{1, \frac{1+\sqrt{d}}{2}\right\}$

The second case requires $\frac{1+\sqrt{d}}{2}$ because it satisfies the monic polynomial $x^2 - x + \frac{1-d}{4} = 0$ .

TheoremPrimitive Element Theorem

Every number field $K$ can be written as $K = \mathbb{Q}(\alpha)$ for some $\alpha \in K$ . However, $\mathcal{O}_K$ need not equal $\mathbb{Z}[\alpha]$ for any single $\alpha$ .

A number field is monogenic if $\mathcal{O}_K = \mathbb{Z}[\alpha]$ for some $\alpha$ . Not all number fields are monogenic; the obstruction is measured by the conductor.

Remark

The failure of monogeneity is a subtle phenomenon. For instance, $K = \mathbb{Q}(\alpha)$ where $\alpha^3 - \alpha^2 - 2\alpha - 8 = 0$ has $\mathcal{O}_K \neq \mathbb{Z}[\alpha]$ . Computing $\mathcal{O}_K$ requires finding all algebraic integers in $K$ , which may involve solving systems of polynomial equations.

The structure theorem guarantees that $\mathcal{O}_K$ is always a Dedekind domain: a Noetherian, integrally closed domain of dimension 1. This algebraic property ensures that every nonzero ideal factors uniquely into prime ideals, recovering unique factorization at the level of ideals.