# AlgebraicIntegerQ

yields True if a is an algebraic integer, and yields False otherwise.

# Details • AlgebraicIntegerQ is typically used to test whether a number is an algebraic integer or not.
• An algebraic integer is a number that is a root of a polynomial with integer coefficients and leading coefficient 1.
• • returns False unless a is manifestly an algebraic integer.

# Examples

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## Basic Examples(2)

Test whether a number is an algebraic integer:

The number is not an algebraic integer:

## Scope(4)

AlgebraicIntegerQ works over integers:

Real numbers:

Complex numbers:

Transcendental numbers:

Root objects:

AlgebraicNumber objects:

## Applications(10)

### Basic Applications(2)

Generate random algebraic integers:

Plot algebraic integers:

Plot the field of at most -degree algebraic integers in the complex plane:

### Special Sequences(3)

Gaussian integers are complex numbers of the form where a and b are integers:

Gaussian integers are algebraic integers:

Plot Gaussian primes:

Eisenstein integers are complex numbers of the form where a and b are integers and ω is the cube root of unity :

Eisenstein integers are algebraic integers:

Check whether an Eisenstein integer is prime:

Plot Eisenstein primes:

A Pisot number is a positive algebraic integer greater than 1, all of whose conjugate elements have absolute value less than 1 [more info]:

### Number Theory(5)

For every rational number q there exists a nonzero integer n such that is an algebraic integer:

All roots of unity are algebraic integers:

Quadratic integers are algebraic integers of degree two:

The only integers that are both algebraic integers and algebraic units are and :

Use the roots of unity to find Cyclotomic polynomials:

## Properties & Relations(8)

The sum and product of two algebraic integers are algebraic integers:

An algebraic integer raised to a rational power is an algebraic integer:

Algebraics represents the domain of all algebraic numbers, including algebraic integers:

An algebraic unit is a number for which both it and its reciprocal are algebraic integers.

The roots of monic polynomials with integer coefficients are algebraic integers:

Use MinimalPolynomial to find the minimal polynomial of an algebraic integer:

Its roots are all algebraic integers:

Use NumberFieldIntegralBasis to get the integral basis for a number field:

Any integer linear combination will be an algebraic integer:

Fundamental units of a number field are algebraic integers:

## Possible Issues(1)

In some cases it is not known whether the number is an algebraic integer: