matrix representation of Frobenius map

Question

I am in an urgent need to know what is the matrix representation of Frobenius map for finie field like GF(4). Let suppose the basis of GF(4) be {1, a+1}. We know that the frobenius map is generator of Galois group, but still I am a little bit confused about its matrix. I would be thankful if you give me a clear answer? I want to know what is the generator of Galois group of GF(4) in a matrix form?

Answer 1

If $F$ is the Frobenius automorphism, then the action on your basis loooks like

• $F(1)=1^2=1\cdot1+0\cdot(a+1)$,
• $F(a+1)=(a+1)^2=a^2+1=(a+1)+1=a=1\cdot1+1\cdot(a+1)$.

You can read the matrix from this the same way you would read the matrix of any linear transformation from a finite dimensional vector space to itself.