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.
Examplesopen allclose all
Basic Examples (2)
AlgebraicIntegerQ works over integers:
AlgebraicIntegerQ threads over lists:
Basic Applications (2)
Special Sequences (3)
Gaussian integers are complex numbers of the form where a and b are integers:
Gaussian integers are algebraic integers:
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:
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:
Wolfram Research (2007), AlgebraicIntegerQ, Wolfram Language function, https://reference.wolfram.com/language/ref/AlgebraicIntegerQ.html.
Wolfram Language. 2007. "AlgebraicIntegerQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AlgebraicIntegerQ.html.
Wolfram Language. (2007). AlgebraicIntegerQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AlgebraicIntegerQ.html