WOLFRAM

Wolfram Language & System Documentation Center

PrimitivePolynomialQ[poly,p]

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

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Related Guides
History
Cite this Page

PrimitivePolynomialQ

PrimitivePolynomialQ[poly,p]

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

Details

  • The polynomial poly must be univariate.

Examples

open all close all

Basic Examples  (2)

Test whether a polynomial is primitive modulo 13:

Wolfram Language code: PrimitivePolynomialQ[x ^ 4 + 3x ^ 3 + 2x ^ 2 + x + 7, 13]

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

Wolfram Language code: PrimitivePolynomialQ[x ^ 2 + 1, 2]

Scope  (3)

Test for primitivity of a univariate polynomial modulo a prime:

Wolfram Language code: PrimitivePolynomialQ[x ^ 2 + 5x - 1, 7]

Polynomials can be given in non-expanded form:

Wolfram Language code: PrimitivePolynomialQ[(x + 1)(x + 3) + 2, 7]

Coefficients of the polynomial do not have to be integers:

Wolfram Language code: PrimitivePolynomialQ[x ^ 2 / 2 + 2x + 5 / 2, 7]

Properties & Relations  (4)

A polynomial must be irreducible in order to be primitive:

Wolfram Language code: IrreduciblePolynomialQ[x ^ 4 + 3x ^ 3 + 2x ^ 2 + x + 7, Modulus -> 13]

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

Wolfram Language code: IrreduciblePolynomialQ[x ^ 4 + 3x ^ 3 + 2x ^ 2 + x + 9, Modulus -> 13]
Wolfram Language code: PrimitivePolynomialQ[x ^ 4 + 3x ^ 3 + 2x ^ 2 + x + 9, 13]

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

Wolfram Language code: MersennePrimeExponent[10]
Wolfram Language code: poly = x ^ 89 + x ^ 38 + 1;
Wolfram Language code: PrimitivePolynomialQ[poly, 2]
Wolfram Language code: IrreduciblePolynomialQ[poly, Modulus -> 2]

Primitivity of a polynomial depends on the choice of prime:

Wolfram Language code: poly = x ^ 4 + 3x ^ 3 + 2x ^ 2 + x + 7;
Wolfram Language code: PrimitivePolynomialQ[poly, 17]
Wolfram Language code: PrimitivePolynomialQ[poly, 23]
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_2026_primitivepolynomialq, author="Wolfram Research", title="{PrimitivePolynomialQ}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_primitivepolynomialq, organization={Wolfram Research}, title={PrimitivePolynomialQ}, year={2017}, url={https://reference.wolfram.com/language/ref/PrimitivePolynomialQ.html}, note=[Accessed: 13-June-2026]}