IrreduciblePolynomialQ

IrreduciblePolynomialQ[poly]

tests whether poly is an irreducible polynomial over the rationals.

IrreduciblePolynomialQ[poly,Modulusp]

tests whether poly is irreducible modulo a prime p.

IrreduciblePolynomialQ[poly,Extension{a1,a2,}]

tests whether poly is irreducible over the field extension generated by the algebraic numbers ai.

IrreduciblePolynomialQ[poly,ExtensionAll]

tests whether poly is absolutely irreducible over the complex numbers.

Details and Options

Examples

open allclose all

Basic Examples  (1)

Test irreducibility of polynomials:

Scope  (8)

Irreducibility of univariate polynomials over the rationals:

Irreducibility of multivariate polynomials over the rationals:

Irreducibility over the Gaussian rationals:

Irreducibility of univariate polynomials over the integers modulo 2:

Irreducibility of multivariate polynomials over the integers modulo 3:

Irreducibility testing modulo a prime may fail for some multivariate polynomials:

By default, algebraic number coefficients are treated as independent variables:

This tests irreducibility over the rationals extended by the algebraic number coefficients:

Irreducibility over a finite algebraic extension of rationals:

Absolute irreducibility over the complex numbers:

Options  (5)

Extension  (3)

By default, algebraic number coefficients are treated as independent variables:

Extension->Automatic automatically extends to a field that covers the coefficients:

The polynomial is irreducible over the rationals:

The same polynomial is reducible over the rationals extended by I and Sqrt[2]:

Absolute irreducibility:

GaussianIntegers  (1)

The polynomial is irreducible over the rationals:

The same polynomial is reducible over the Gaussian rationals:

Modulus  (1)

Irreducibility modulo a prime:

Properties & Relations  (2)

A polynomial is irreducible if FactorList gives one nonconstant factor with exponent 1:

IrreduciblePolynomialQ may be significantly faster than FactorList:

Possible Issues  (1)

Irreducibility testing modulo a prime may fail for some multivariate polynomials:

Wolfram Research (2008), IrreduciblePolynomialQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html.

Text

Wolfram Research (2008), IrreduciblePolynomialQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html.

BibTeX

@misc{reference.wolfram_2021_irreduciblepolynomialq, author="Wolfram Research", title="{IrreduciblePolynomialQ}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html}", note=[Accessed: 24-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_irreduciblepolynomialq, organization={Wolfram Research}, title={IrreduciblePolynomialQ}, year={2008}, url={https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html}, note=[Accessed: 24-September-2021 ]}

CMS

Wolfram Language. 2008. "IrreduciblePolynomialQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html.

APA

Wolfram Language. (2008). IrreduciblePolynomialQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html