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
open all close allBasic Examples (1)
Scope (6)
Compute PolynomialGCD over the integers modulo 2:
PolynomialGCD[x ^ 3 + x ^ 2 + x + 1, (x + 1) ^ 4, Modulus -> 2]Factor a polynomial over the integers modulo 3:
Factor[x ^ 3 + 2x ^ 2 + 3x + 5, Modulus -> 3]Find a GroebnerBasis over the integers modulo 5:
GroebnerBasis[{x ^ 3 - 2x y + y ^ 2, y ^ 3 - x ^ 2y + 3}, {x, y}, Modulus -> 5]Reduce equations over the integers modulo 7:
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:
Det[{{1, 2, 3}, {21, 210, 2100}, {2221, 3332, 4443}}, Modulus -> 8]Find a modulus for which a system of equations has a solution:
Solve[x ^ 2 + y ^ 2 == 1 && 2x + 3y == 5 && x ^ 3 - x y == 7, {x, y}, Modulus -> Automatic]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