Difference between Galois and other automorphisms

Question

With respect to the definition of Galois field, for $E$ an extension of $F$ ($E$ and $F$ are finite fields) $\mathrm{Gal}\,(E/F)$ is the set of automorphisms of $E$ which fix $F$ pointwise. So I think that we can distinguish between $\mathrm{Gal}\,(E/F)$ and $\mathrm{Aut}\,(E)$, right?

Answer 1

You don't need to assume that $E$ and $F$ are finite fields. Every automorphism of a field fixes its prime field, by which I mean $\mathbb Q$ or $\mathbb F_p$, the field generated by $1\in E$. However it doesn't need to fix any other subfield, so that's what makes the Galois group different.