## You are here

Homealgebraic integer

## Primary tabs

# algebraic integer

*algebraic integer* of $K$ if it is the root of a monic polynomial with coefficients in $\mathbb{Z}$, i.e., an element of $K$ that is integral over $\mathbb{Z}$. Every algebraic integer is an algebraic number (with $K=\mathbb{C}$), but the converse is false.

Keywords:

algebraic number theory

Related:

IntegralBasis, CyclotomicUnitsAreAlgebraicUnits, FundamentalUnits, Monic2, RingWithoutIrreducibles

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11R04*no label found*62-01

*no label found*03-01

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## may wish to generalize

You might want to add the simple generalization to an extension of the rationals that isn't all of C:

Let K be an extension of Q. A number $\alpha \in K$ is called an algebraic integer of K it is the root of a monic polynomial with coefficients in $\mathbb{Z}$.

## Re: may wish to generalize

Good idea, thanks.