Ring of Integers of Number Field is Free Z-Module

Theorem

Let $K$ be an algebraic number field.

Let $\OO_K$ be its ring of integers.

Let $\sqbrk {K : \Q}$ denote the degree of field extension $K : \Q$.

Then $\OO_K$ is a free $\Z$-module of dimension $\sqbrk {K : \Q}$.

Proof

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Sources

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• 2011: J.S. Milne: Algebraic Number Theory: Chapter $2$ Ring of Integers: Rings of integers are finitely generated: Proposition $2.29$