.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.
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.