Definition:Algebraic Integer

Definition
Let $K / \Q$ be an algebraic number field.

Then $\alpha \in K$ is an algebraic integer if it satisfies a monic polynomial $f \in \Z \sqbrk X$.

The set of all algebraic integers in $K$ is denoted $\mathfrak o_K$ or $\OO_K$.

By Ring of Algebraic Integers it is a ring, hence usually referred to as the ring of algebraic integers of $K$.

Also see

 * Definition:Ring of Algebraic Integers
 * Definition:Integral Closure