# Definition:Algebraic Integer

## Definition

Let $K / \Q$ be a number field.

Then $\alpha \in K$ is an algebraic integer if it satisfies a monic polynomial $f \in \Z \left[{X}\right]$.

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

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

Let $K / \Q$ be a number field.
Let $K / \Q$ have degree two.
Then an algebraic integer in $K$ is a quadratic integer.