Ring of integers of a number field

Let $𝐾$ be a number field. Then the ring of integers $O 𝐾$ is a Dedekind domain. #m/thm/ring

Proof

First we show $O 𝐾$ to be a Noetherian ring. Let $𝐼 1 ⊴ 𝐼 2 ⊴ \dots$ be an increasing sequence of ideals, and without loss of generality take $𝐼 1 \neq 0$ . By The ring of integers of a number field forms a lattice, it follows $O 𝐾 / 𝐼 1$ is finite, implying there are only finitely many subrings of $O 𝐾$ containing $𝐼 1$ and thus the sequence must stabilize. Therefore $O 𝐾$ is Noetherian.

Now let $𝔭 ◃ O 𝐾$ be a nonzero prime ideal. It follows from ^C1 that $O / 𝔭$ is finite, and A finite integral domain is a field, thus $𝔭$ is maximal since $𝑅 / 𝐼$ for commutative $𝑅$ is a field iff $𝐼$ is maximal. Therefore, $O 𝐾$ has Krull dimension $d i m O 𝐾 = 1$ .

Since the ring of integers is automatically integrally closed, it follows $O 𝐾$ is Dedekind.

Further terminology

Properties

#state/tidy | #lang/en | #SemBr