Definition:Ring of Integers of Number Field

Definition

Let $K$ be an number field.

The ring of integers of $K$, usually denoted $\OO_K$ or $\mathfrak o_K$, is the integral closure of $\Z$ in $K$.

Also defined as

Some sources define a ring of integers specifically on an algebraic number field.

The motivation behind this decision is that some mathematicians are not very interested in the general case.

However, this point of view goes against the inclusivist philosophical position endorsed by $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

• Ring of Integers of Number Field is Free Z-Module
• Ring of Integers of Number Field is Dedekind Domain

• Results about rings of integers of number fields can be found here.

Special cases

• Definition:Ring of Gaussian Integers
• Definition:Ring of Eisenstein Integers

Sources

• 1991: Jürgen Neukirch: Algebraic Number Theory: Chapter $\text I$: Algebraic Integers: $\S 2$ Integrality