# 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 allclose all

## Basic Examples(1)

Solve equations:

Factor polynomials:

Compute inverse:

## Scope(6)

Compute PolynomialGCD over the integers modulo 2:

Factor a polynomial over the integers modulo 3:

Find a GroebnerBasis over the integers modulo 5:

Reduce equations over the integers modulo 7:

Compute the determinant of a matrix modulo 8:

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

## Properties & Relations(2)

Factor a polynomial over a finite field:

Factor a polynomial over a finite Extension of rationals: