Number Fields and Rings of Integers - Core Definitions

Number fields form the foundation of algebraic number theory, generalizing the rational numbers $\mathbb{Q}$ to finite-degree field extensions that preserve algebraic structure.

DefinitionNumber Field

A number field is a finite-degree field extension $K$ of the rational numbers $\mathbb{Q}$ . That is, $K$ is a field containing $\mathbb{Q}$ such that $[K : \mathbb{Q}] = n < \infty$ .

Every element $\alpha \in K$ is algebraic over $\mathbb{Q}$ , meaning it satisfies a polynomial equation with rational coefficients. The degree $[K : \mathbb{Q}]$ is called the degree of the number field.

ExampleClassical Number Fields

Quadratic fields: $K = \mathbb{Q}(\sqrt{d})$ where $d \in \mathbb{Z}$ is square-free, with degree $[K : \mathbb{Q}] = 2$
Cyclotomic fields: $K = \mathbb{Q}(\zeta_n)$ where $\zeta_n = e^{2\pi i/n}$ is a primitive $n$ -th root of unity
Cubic fields: Extensions of degree 3, such as $\mathbb{Q}(\sqrt[3]{2})$
The rationals themselves: $\mathbb{Q}$ is the unique number field of degree 1

DefinitionRing of Integers

Let $K$ be a number field. An element $\alpha \in K$ is called an algebraic integer if it is a root of a monic polynomial with integer coefficients. The set of all algebraic integers in $K$ is denoted $\mathcal{O}_K$ and called the ring of integers of $K$ .

The ring $\mathcal{O}_K$ satisfies:

$\mathcal{O}_K$ is a ring containing $\mathbb{Z}$
$\mathcal{O}_K$ is finitely generated as a $\mathbb{Z}$ -module
The field of fractions of $\mathcal{O}_K$ is $K$

The ring of integers $\mathcal{O}_K$ plays a central role analogous to $\mathbb{Z}$ in $\mathbb{Q}$ . However, unlike $\mathbb{Z}$ , the ring $\mathcal{O}_K$ may fail to be a unique factorization domain. For example, in $\mathbb{Q}(\sqrt{-5})$ , we have $6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$ , giving two distinct factorizations.

ExampleRings of Integers

Gaussian integers: $\mathcal{O}_{\mathbb{Q}(i)} = \mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}$
Eisenstein integers: $\mathcal{O}_{\mathbb{Q}(\omega)} = \mathbb{Z}[\omega]$ where $\omega = e^{2\pi i/3}$
Quadratic fields: For $K = \mathbb{Q}(\sqrt{d})$ with $d$ square-free, $\mathbb{Z}[\sqrt{d}] & \text{if } d \equiv 2, 3 \pmod{4} \\ \mathbb{Z}\left[\frac{1 + \sqrt{d}}{2}\right] & \text{if } d \equiv 1 \pmod{4} \end{cases}$$$

Remark

The failure of unique factorization in $\mathcal{O}_K$ motivates the theory of ideals and Dedekind domains, where unique factorization is restored at the level of ideals rather than elements.

Understanding the structure of $\mathcal{O}_K$ as a $\mathbb{Z}$ -module is fundamental. The ring $\mathcal{O}_K$ always admits an integral basis: a basis $\{\omega_1, \ldots, \omega_n\}$ as a $\mathbb{Z}$ -module, so every element of $\mathcal{O}_K$ can be uniquely written as $a_1\omega_1 + \cdots + a_n\omega_n$ with $a_i \in \mathbb{Z}$ .