Let

be a finite field of characteristic

As such it is a finite-dimensional vector space over

.

a) Prove that the Frobenius morphism

is a linear map over

b) Prove that the geometric multiplicity of 1 as an eigenvalue of

is 1.

This means you have to prove the dimension of the eigenspace corresponding to 1 is

1, but , so...
c) Let

have dimension 2 over

Prove that 2 is not an eigenvalue of

.

2 is an eigenvalue iff , which

of course is impossible.

Tonio
I solved a) using Fermat's Little Theorem, but am unsure about b) and c). I think I solved c) by considering

, but I'm not sure if there's a simpler proof.

Thanks!