Definition:Algebraic Integer

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

We call $\alpha \in K$ an algebraic integer if it satisfies a monic polynomial $f \in \Z[X]$.

The set of all algebraic integers is usually 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 $K$.

Quadratic Integers
If $K/\Q$ has degree two, then an algebraic integer in $K$ is called a quadratic integer.

Examples incluse the Gaussian integers and Eisenstein integers.