IrreduciblePolynomialQ

tests whether poly is an irreducible polynomial over the rationals.

tests whether poly is irreducible modulo a prime p.

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

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.

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Test irreducibility of polynomials:

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:

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:

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:

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