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PolynomialMod[poly,m]

gives the polynomial poly reduced modulo m.

PolynomialMod[poly,{m1,m2,…}]

reduces modulo all of the mi.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
CoefficientDomain  
Modulus  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
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History
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PolynomialMod

PolynomialMod[poly,m]

gives the polynomial poly reduced modulo m.

PolynomialMod[poly,{m1,m2,…}]

reduces modulo all of the mi.

Details and Options

  • PolynomialMod[poly,m] for integer m gives a polynomial in which all coefficients are reduced modulo m.
  • When m is a polynomial, PolynomialMod[poly,m] reduces poly by subtracting polynomial multiples of m, to give a result with minimal degree and leading coefficient.
  • PolynomialMod gives results according to a definite convention; other conventions could yield results differing by multiples of m.
  • Unlike PolynomialRemainder, PolynomialMod never performs divisions in generating its results.
  • PolynomialMod takes the following options:
  • CoefficientDomain the type of objects assumed to be coefficients
    Modulus 0modulus to assume for integers
    TrigFalsewhether to do trigonometric as well as algebraic transformations

Examples

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Basic Examples  (2)

Reduce a polynomial modulo 2:

Reduce a polynomial modulo another polynomial:

Scope  (6)

Reduce a polynomial modulo an integer:

Reduce a multivariate polynomial modulo an integer:

Reduce a polynomial modulo a polynomial:

The difference of the original polynomial and the result is divisible by the modulus:

Reduce a polynomial modulo a polynomial with complex coefficients:

Reduce a polynomial modulo a polynomial and an integer:

Reduce a polynomial modulo two polynomials and an integer:

Options  (3)

CoefficientDomain  (2)

With the default CoefficientDomain->Rationals, integer coefficients can be inverted:

With CoefficientDomain->Integers, PolynomialMod does not invert integer coefficients:

Modulus  (1)

Reduce a polynomial modulo a polynomial over the integers modulo 3:

Applications  (1)

Reduce all coefficients of a polynomial modulo an integer:

Properties & Relations  (5)

For univariate rational polynomials, PolynomialRemainder is the same as PolynomialMod:

PolynomialRemainder considers all polynomials to be univariate in the specified variable:

For multivariate polynomials, PolynomialMod picks its own variable order:

The main variable here is a:

PolynomialRemainder considers parameters to be invertible:

PolynomialMod does not invert symbolic expressions:

Wolfram Research (1991), PolynomialMod, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialMod.html (updated 2003).

Text

Wolfram Research (1991), PolynomialMod, Wolfram Language function, https://reference.wolfram.com/language/ref/PolynomialMod.html (updated 2003).

CMS

Wolfram Language. 1991. "PolynomialMod." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/PolynomialMod.html.

APA

Wolfram Language. (1991). PolynomialMod. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PolynomialMod.html

BibTeX

@misc{reference.wolfram_2025_polynomialmod, author="Wolfram Research", title="{PolynomialMod}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/PolynomialMod.html}", note=[Accessed: 01-March-2026]}

BibLaTeX

@online{reference.wolfram_2025_polynomialmod, organization={Wolfram Research}, title={PolynomialMod}, year={2003}, url={https://reference.wolfram.com/language/ref/PolynomialMod.html}, note=[Accessed: 01-March-2026]}