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
- The polynomial poly can involve any number of variables.
- IrreduciblePolynomialQ[poly,GaussianIntegers->True] tests whether poly is irreducible over the Gaussian rationals.
- If any coefficients in poly are complex numbers, irreducibility testing is done over the Gaussian rationals.
- With the default setting Extension->None, IrreduciblePolynomialQ[poly] will treat algebraic number coefficients in poly like independent variables.
- IrreduciblePolynomialQ[poly,Extension->Automatic] extends the domain of coefficients to include any algebraic numbers that appear in poly.
- IrreduciblePolynomialQ automatically threads over lists.
Examples
open allclose allScope (9)
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 over a finite field:
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:
Options (7)
Extension (5)
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]:
The polynomial 
is irreducible over 
:
The same polynomial is reducible over a degree-
extension of 
:
The polynomial 
is irreducible over the finite field 
with 
elements:
becomes reducible after embedding 
in a finite field 
with 
elements:
GaussianIntegers (1)
Properties & Relations (1)
A polynomial is irreducible if FactorList gives one nonconstant factor with exponent 1:
Text
Wolfram Research (2008), IrreduciblePolynomialQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html (updated 2023).
CMS
Wolfram Language. 2008. "IrreduciblePolynomialQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. 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