Ring of integers

Ring of integers of a number field

Let be a number field. Then the ring of integers is a Dedekind domain. ring

Proof

First we show to be a Noetherian ring. Let be an increasing sequence of ideals, and without loss of generality take . By The ring of integers of a number field forms a lattice, it follows is finite, implying there are only finitely many subrings of containing and thus the sequence must stabilize. Therefore is Noetherian.

Now let be a nonzero prime ideal. It follows from ^C1 that is finite, and A finite integral domain is a field, thus is maximal since Condition for a quotient commutative ring to be a field. Therefore, has Krull dimension .

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

Further terminology

• Absolute norm of an ideal of the ring of integers of a number field
• Splitting of prime ideals in a number field

Properties

• Ideal class group of a number field
• The ring of integers of a number field forms a lattice

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