Polynomials Modulo Primes

The Wolfram Language 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.
In[1]:=
Click for copyable input
Out[1]=
This reduces the coefficients modulo 2.
In[2]:=
Click for copyable input
Out[2]=
Here are the factors of the resulting polynomial over the integers.
In[3]:=
Click for copyable input
Out[3]=
If you work modulo 2, further factoring becomes possible.
In[4]:=
Click for copyable input
Out[4]=