# 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

open allclose all

## Basic Examples(1)

Test irreducibility of polynomials:

## Scope(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:

Absolute irreducibility over the complex numbers:

## Options(7)

### Extension(5)

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[2]:

Absolute irreducibility:

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)

The polynomial is irreducible over the rationals:

The same polynomial is reducible over the Gaussian rationals:

### Modulus(1)

Irreducibility modulo a prime:

## Properties & Relations(1)

A polynomial is irreducible if FactorList gives one nonconstant factor with exponent 1:

Wolfram Research (2008), IrreduciblePolynomialQ, Wolfram Language function, https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html (updated 2023).

#### 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

#### BibTeX

@misc{reference.wolfram_2024_irreduciblepolynomialq, author="Wolfram Research", title="{IrreduciblePolynomialQ}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html}", note=[Accessed: 12-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_irreduciblepolynomialq, organization={Wolfram Research}, title={IrreduciblePolynomialQ}, year={2023}, url={https://reference.wolfram.com/language/ref/IrreduciblePolynomialQ.html}, note=[Accessed: 12-August-2024 ]}