Definition:Algebraic Integer

Definition

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

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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.

Also see

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

• Results about algebraic integers can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): algebraic integer
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebraic integer