Definition:Algebraic Integer

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Definition

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

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

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

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

Quadratic Integer

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

Let $K / \Q$ have degree two.

Then an algebraic integer in $K$ is a quadratic integer.