automorphism of fields

Question

let we consider $GF(p^n)$ as a vector space over $GF(p)$, $p$ is prime. Also we want to have an invertible linear map on $GF(p^n)$, (automorphism of $GF(p^n)$). on the other hand we know that A field automorphism is a bijective ring homomorphism from a field to itself. I want to know what are the differences between these two automorphism? Can we consider them the same?

Answer 1

For example, let $a$ has a minimal polynomial of degree 2 ove $F=GF(p)$. Then $E=F(a)=\{ k \cdot 1 + l \cdot a | k,l \in F\}$. For any field automorphism $\alpha$ of $E$, $\alpha(1)=1$. But $1 \rightarrow a, a\rightarrow 1$ will give us an invertible linear transformation.