# Definition:Algebraic Integer

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## Definition

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

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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 $K$.

### 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**.

## Also see

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
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