# Parametric solution of a system of polynomial equations

I have the following system of equations,

1+x+y+z==0, 1+x*y+y*z+x*z==0


which I want to solve in the extension field of GF(2), the algebraic closure of GF(2) for example. There is a parametric solution of these equations in terms of the parameter s as x=1+s, y=1+$$\omega$$ s, z=1+$$\omega^2$$s where $$s$$ is the parameter and $$\omega^2+\omega+1=0$$. Is there a way for Mathematica to give such parametric solutions to an incomplete system of polynomial equations?

If you substitude {x -> 1 + s, y -> 1 + \[Omega] s , z -> 1 + \[Omega]^2 s} into your equation you can solve for s,\[Omega]
eqn = {1 + x + y + z == 0 && 1 + x*y + y*z + x*z == 0}

• @cleanplay My result shows that your suggested parametric solution is only valid for special values s, \[Omega] – Ulrich Neumann Apr 9 '19 at 10:52
• I think that is because you chose $\omega$ to have solution in reals or complex numbers, while it is an element of $GF(4)$ – cleanplay Apr 10 '19 at 1:38