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PrimitivePolynomialQ[poly,p]

tests whether poly is a primitive polynomial modulo a prime p.

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Basic Examples  
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PrimitivePolynomialQ

PrimitivePolynomialQ[poly,p]

tests whether poly is a primitive polynomial modulo a prime p.

Details

  • The polynomial poly must be univariate.

Examples

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Basic Examples  (2)

Test whether a polynomial is primitive modulo 13:

This polynomial can be factored modulo 2, and therefore it is not primitive:

Scope  (3)

Test for primitivity of a univariate polynomial modulo a prime:

Polynomials can be given in non-expanded form:

Coefficients of the polynomial do not have to be integers:

Properties & Relations  (4)

A polynomial must be irreducible in order to be primitive:

Irreducibility is a necessary but not-sufficient condition for a polynomial to be primitive:

A trinomial whose order is a Mersenne prime exponent is primitive modulo 2 if and only if it is irreducible:

Primitivity of a polynomial depends on the choice of prime:

Wolfram Research (2017), PrimitivePolynomialQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html.

Text

Wolfram Research (2017), PrimitivePolynomialQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html.

CMS

Wolfram Language. 2017. "PrimitivePolynomialQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html.

APA

Wolfram Language. (2017). PrimitivePolynomialQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html

BibTeX

@misc{reference.wolfram_2025_primitivepolynomialq, author="Wolfram Research", title="{PrimitivePolynomialQ}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html}", note=[Accessed: 13-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_primitivepolynomialq, organization={Wolfram Research}, title={PrimitivePolynomialQ}, year={2017}, url={https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html}, note=[Accessed: 13-August-2025]}