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asked 6 years ago

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solution of a system of equations in algebraic closure of GF2

How do I look for solutions of a system of equations in a particular field? For example, the following set of equations in variables a,b,c,d,e,f 1+a+c+e=0,b+d+f=0,1+ae+ce+ac=0,be+af+cf+ed+bc+ad=0,bf+fd+bd=0 have the solution a=1,b=1,c=1,d=ω,e=1,f=ω2 where ω is the 3rd root of unity. This lies in an extension of GF2, GF4=GF2[u]u2+u+1 where GF2[u] is a polynomial ring with variable u and one representation of GF4 is 0,1,ω,ω2.

I have a set of equations and I want to know whether there exist solutions of these equations in an extension of Galois field GF2 and what are they? Is there a way to check this in Sage?