This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# IrreduciblePolynomialQ

 IrreduciblePolynomialQ[poly] tests whether poly is an irreducible polynomial over the rationals. IrreduciblePolynomialQ[poly, Modulus->p]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 . IrreduciblePolynomialQ[poly, Extension->All]tests whether poly is absolutely irreducible over the complex numbers.
• The polynomial poly can involve any number of variables.
• If any coefficients in poly are complex numbers, irreducibility testing is done over the Gaussian rationals.
Test irreducibility of polynomials:
Test irreducibility of polynomials:
 Out[1]=
 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 :
Irreducibility of multivariate polynomials over the integers modulo :
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:
The currently implemented methods may not be sufficient for some polynomials:
 Options   (5)
By default, algebraic number coefficients are treated as independent variables:
Extension 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:
Absolute irreducibility:
The polynomial is irreducible over the rationals:
The same polynomial is reducible over the Gaussian rationals:
Irreducibility modulo a prime:
A polynomial is irreducible if FactorList gives one nonconstant factor with exponent 1:
IrreduciblePolynomialQ may be significantly faster than FactorList:
Irreducibility testing modulo a prime may fail for some multivariate polynomials:
Absolute irreducibility testing may fail for some multivariate polynomials:
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