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


  • 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.
  • AlgebraicIntegerQ[a] returns False unless a is manifestly an algebraic integer.


open allclose all

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:

Radical expressions:

Root objects:

AlgebraicNumber objects:

AlgebraicIntegerQ threads over lists:

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:

Wolfram Research (2007), AlgebraicIntegerQ, Wolfram Language function,


Wolfram Research (2007), AlgebraicIntegerQ, Wolfram Language function,


Wolfram Language. 2007. "AlgebraicIntegerQ." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). AlgebraicIntegerQ. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_algebraicintegerq, author="Wolfram Research", title="{AlgebraicIntegerQ}", year="2007", howpublished="\url{}", note=[Accessed: 17-June-2024 ]}


@online{reference.wolfram_2024_algebraicintegerq, organization={Wolfram Research}, title={AlgebraicIntegerQ}, year={2007}, url={}, note=[Accessed: 17-June-2024 ]}