Computation in a ring and Galois Field

[parisse]

You can not build a finite field with a polynomial that is not irreducible. But that's not the right way to invert a polynomial modulo another one, the right way is a call to the extended gcd algorithm. The commandname is egcd.
For example if a:=x^3 + 1 %% 2and b:=x^5+x^3+1 %% 2, egcd(a,b) will return three polynomials u, v and d such that d is the gcd of the input polynomials and a*u+b*v=d. If a and b are coprime, d will be 1 and modulo b you have a*u==1, therefore the inverse of a is u.