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Polynomials Modulo Primes

Mathematica can work with polynomials whose coefficients are in the finite field of integers modulo a prime .

PolynomialMod[poly,p]reduce the coefficients in a polynomial modulo p
Expand[poly,Modulus->p]expand poly modulo p
Factor[poly,Modulus->p]factor poly modulo p
PolynomialGCD[poly1,poly2,Modulus->p]
find the GCD of the modulo p
GroebnerBasis[polys,vars,Modulus->p]
find the Gröbner basis modulo p

Functions for manipulating polynomials over finite fields.

Here is an ordinary polynomial.
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This reduces the coefficients modulo 2.
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Here are the factors of the resulting polynomial over the integers.
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If you work modulo 2, further factoring becomes possible.
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