Theorem 1   Let $K$ be a number field of degree $n$ over $\Rats$ and let $\mathcal{O}_K$ be the ring of integers of $K$ . The ring $\mathcal{O}_K$ 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 $\alpha_1,\ldots,\ \alpha_n$ such that every element of $\mathcal{O}_K$ can be expressed uniquely as a $\Ints$ -linear combination of the $\alpha_i$ .

Corollary 1   Every ideal of $\mathcal{O}_K$ is finitely generated.

Proof. [Proof of the corollary] By the theorem, $\mathcal{O}_K$ 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.