the ring of integers of a number field is finitely generated over $\mathbb{Z}$

Theorem.

Let $K$ be a number field of degree $n$ over $\mathrm{Q}$ and let ${\mathrm{O}}_K$ be the ring of integers of $K$. The ring ${\mathrm{O}}_K$ is a free abelian group of rank $n$. In other words, there exists a finite integral basis (with $n$ elements) for $K$, i.e. there exist algebraic integers ${\alpha }_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{\alpha }_n$ such that every element of ${\mathrm{O}}_K$ can be expressed uniquely as a $\mathrm{Z}$-linear combination of the ${\alpha }_i$.

Corollary.

Every ideal of ${\mathrm{O}}_K$ is finitely generated.

Proof of the corollary.

By the theorem, ${𝒪}_K$ is a free abelian group of rank $n$, and therefore it is finitely generated. Notice that an ideal is an additive subgroup. Finally a subgroup of a finitely generated free abelian group is also finitely generated.∎

This is the first step to prove that ${𝒪}_K$ is a Dedekind domain. Notice that the field of fractions of ${𝒪}_K$ is the field $K$ itself. Therefore, by definition, ${𝒪}_K$ is integrally closed in $K$.