Built-in Mathematica Symbol

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 a_i.

IrreduciblePolynomialQ[poly, Extension->All]
tests whether poly is absolutely irreducible over the complex numbers.

MORE INFORMATION

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

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Basic Examples (1)

Test irreducibility of polynomials:

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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:

The currently implemented methods may not be sufficient for some polynomials:

Options (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]:

Absolute irreducibility:

The polynomial

is irreducible over the rationals:

The same polynomial is reducible over the Gaussian rationals:

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 (2)

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

Absolute irreducibility testing may fail for some multivariate polynomials: