Number Fields and Rings of Integers - Key Properties

The ring of integers $\mathcal{O}_K$ of a number field $K$ possesses rich structural properties that generalize classical results from the integers $\mathbb{Z}$ .

DefinitionDiscriminant

Let $K$ be a number field of degree $n$ with integral basis $\{\omega_1, \ldots, \omega_n\}$ . The discriminant is defined as $\Delta_K = \det(\sigma_i(\omega_j))^2$ where $\sigma_1, \ldots, \sigma_n$ are the $n$ embeddings of $K$ into $\mathbb{C}$ . The discriminant is a nonzero integer independent of the choice of integral basis (up to sign).

The discriminant measures how far $\mathcal{O}_K$ is from being generated by a single element. For quadratic fields $K = \mathbb{Q}(\sqrt{d})$ , the discriminant is $\Delta_K = d$ if $d \equiv 1 \pmod{4}$ and $\Delta_K = 4d$ if $d \equiv 2, 3 \pmod{4}$ .

DefinitionTrace and Norm

For $\alpha \in K$ , the trace and norm are defined using the embeddings $\sigma_1, \ldots, \sigma_n : K \to \mathbb{C}$ : $\text{Tr}_{K/\mathbb{Q}}(\alpha) = \sum_{i=1}^n \sigma_i(\alpha), \quad N_{K/\mathbb{Q}}(\alpha) = \prod_{i=1}^n \sigma_i(\alpha)$

Both the trace and norm map $K$ to $\mathbb{Q}$ , and if $\alpha \in \mathcal{O}_K$ , then $\text{Tr}_{K/\mathbb{Q}}(\alpha), N_{K/\mathbb{Q}}(\alpha) \in \mathbb{Z}$ .

ExampleTrace and Norm in Quadratic Fields

For $K = \mathbb{Q}(\sqrt{d})$ and $\alpha = a + b\sqrt{d}$ with $a, b \in \mathbb{Q}$ : $\text{Tr}_{K/\mathbb{Q}}(\alpha) = 2a, \quad N_{K/\mathbb{Q}}(\alpha) = a^2 - db^2$

The norm form $N(\alpha) = a^2 - db^2$ is a quadratic form that encodes the arithmetic of $K$ .

Remark

The trace and norm are fundamental invariants. The norm is multiplicative: $N(\alpha\beta) = N(\alpha)N(\beta)$ , while the trace is additive: $\text{Tr}(\alpha + \beta) = \text{Tr}(\alpha) + \text{Tr}(\beta)$ . Units in $\mathcal{O}_K$ are precisely the elements with $|N(\alpha)| = 1$ .

A crucial property of $\mathcal{O}_K$ is that it is Noetherian: every ideal is finitely generated. Moreover, $\mathcal{O}_K$ is integrally closed in $K$ , meaning every element of $K$ that is integral over $\mathcal{O}_K$ already lies in $\mathcal{O}_K$ . These properties make $\mathcal{O}_K$ a Dedekind domain, the subject of our next chapter.

The Minkowski bound provides a computational tool: every ideal class of $K$ contains an integral ideal $\mathfrak{a}$ with norm $N(\mathfrak{a}) \leq M_K = \frac{n!}{n^n}\left(\frac{4}{\pi}\right)^{r_2}\sqrt{|\Delta_K|}$ where $r_2$ is the number of pairs of complex embeddings. This bound is essential for computing class numbers.