# 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.

tests whether poly is absolutely irreducible over the complex numbers.

# Details and Options • The polynomial poly can involve any number of variables.
• 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 , IrreduciblePolynomialQ[poly] will treat algebraic number coefficients in poly like independent variables.
• 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:

## 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:

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:

### 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: 