Ring of integers in a number field

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

"ring of integers" redirects here. For the ring of rational integers, see ring of rational integers

Definition

Let $K$ be a number field. Then the ring of integers in $K$, denoted $O_K$, is the subring of $K$ comprising all those elements which are integral over $\mathbb{Z}$, in other words, which satisfy monic polynomials with integer coefficients.

The ring of integers in a number field is also termed a maximal order, here an order is a subring of $K$ which is free as a $\mathbb{Z}$-module, and which, over $\mathbb{Q}$, generates $K$.

Relation with other properties

Weaker properties

• Dedekind domain