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Modulusn

is an option that can be given in certain algebraic functions to specify that integers should be treated modulo n.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
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Modulus

Modulusn

is an option that can be given in certain algebraic functions to specify that integers should be treated modulo n.

Details

  • Modulus appears as an option in Solve, Reduce, Factor, PolynomialGCD, and PolynomialLCM, as well as in linear algebra functions such as Inverse, LinearSolve, and Det.
  • Arithmetic is usually done over the full ring of integers; setting the option Modulus specifies that arithmetic should instead be done in the finite ring .
  • The setting Modulus->0 specifies the full ring of integers.
  • Some functions require that Modulus be set to a prime, or a power of a prime. is a finite field when is prime.
  • Equations for Modulus can be given in Eliminate and related functions.

Examples

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Basic Examples  (1)

Solve equations:

Wolfram Language code: Reduce[x ^ 5 == y ^ 4 + xy + 1, {x, y}, Modulus -> 4]

Factor polynomials:

Wolfram Language code: Table[Factor[x ^ 6 + 3x ^ 4 + x, Modulus -> Prime[k]], {k, 1, 3}]

Compute inverse:

Wolfram Language code: Inverse[{{1, 2}, {3, 4}}, Modulus -> 3]

Scope  (6)

Compute PolynomialGCD over the integers modulo 2:

Wolfram Language code: PolynomialGCD[x ^ 3 + x ^ 2 + x + 1, (x + 1) ^ 4, Modulus -> 2]

Factor a polynomial over the integers modulo 3:

Wolfram Language code: Factor[x ^ 3 + 2x ^ 2 + 3x + 5, Modulus -> 3]

Find a GroebnerBasis over the integers modulo 5:

Wolfram Language code: GroebnerBasis[{x ^ 3 - 2x y + y ^ 2, y ^ 3 - x ^ 2y + 3}, {x, y}, Modulus -> 5]

Reduce equations over the integers modulo 7:

Wolfram Language code: Reduce[x ^ 3 - 3x y + 2y ^ 2 == 0 && y ^ 3 - x ^ 2y + 3 == 0, {x, y}, Modulus -> 7]

Compute the determinant of a matrix modulo 8:

Wolfram Language code: Det[{{1, 2, 3}, {21, 210, 2100}, {2221, 3332, 4443}}, Modulus -> 8]

Find a modulus for which a system of equations has a solution:

Wolfram Language code: Solve[x ^ 2 + y ^ 2 == 1 && 2x + 3y == 5 && x ^ 3 - x y == 7, {x, y}, Modulus -> Automatic]

Properties & Relations  (2)

Factor a polynomial over a finite field:

Wolfram Language code: Factor[x ^ 4 - 2, Modulus -> 3]

Factor a polynomial over a finite Extension of rationals:

Wolfram Language code: Factor[x ^ 4 - 2, Extension -> Sqrt[2]]
Wolfram Research (1988), Modulus, Wolfram Language function, https://reference.wolfram.com/language/ref/Modulus.html.

Text

Wolfram Research (1988), Modulus, Wolfram Language function, https://reference.wolfram.com/language/ref/Modulus.html.

CMS

Wolfram Language. 1988. "Modulus." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Modulus.html.

APA

Wolfram Language. (1988). Modulus. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Modulus.html

BibTeX

@misc{reference.wolfram_2026_modulus, author="Wolfram Research", title="{Modulus}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Modulus.html}", note=[Accessed: 08-October-2026]}

BibLaTeX

@online{reference.wolfram_2026_modulus, organization={Wolfram Research}, title={Modulus}, year={1988}, url={https://reference.wolfram.com/language/ref/Modulus.html}, note=[Accessed: 08-October-2026]}