Automorphisms of Finite Fields

There is a short proof that in GF(p^q) the function φ(x) = x^p is an automorphism. See this. I want to understand the entire group of automorphisms. As a permutation, φ has order q, which is to say that x^{p^q} = x for all x.

If ψ is an automorphism and we know ψ(X) then ψ(x) is determined for all other x.

As you can see this notation is very confusing. I shall attempt to see if lambda notation works. I recommence:

There is a short proof that in GF(p^q) φ = λx.x^p is an automorphism. Within the group of automorphisms, φ has order q for φ^q is the identity; i.e. x^{p^q} = x for all x.

If ψ is an automorphism then ψ(λx.x) entirely determines ψ. Given ψ(λx.x), ψ(y) = ξ