In their treatment of Galois Theory, Dummit and Foote first prove the following result:
Proposition 5. Let $E$ be the splitting field over $F$ of the polynomial $f(x) \in F[x]$. Then $$|\operatorname{Aut}(E/F)| \leq [E:F]$$with equality if $f(x)$ is separable over $F$.
In the next section the following more general result is proven:
Corollary 10. Let $K/F$ be any finite extension. Then $$|\operatorname{Aut}(K/F)| \leq [K:F]$$with equality if and only if $F$ is the fixed field of $\operatorname{Aut}(K/F)$.
I'm trying to see explicitly how Prop. 5 is a special case of Cor. 10 by convincing myself that if $f(x)$ is separable over $F$, then $F$ is the fixed field of $\operatorname{Aut}(E/F)$; but I'm having trouble seeing the connection. Any ideas?
Best Answer
There is a way to see this via Fundamental theorem of Galois theory which is proved later in Dummit Foote's book. So one notes $E/F$ is a Galois extension under this condition, and this gives one-to-one correspondence between subfields of $K$ containing $F$ and subgroups of $\text{Gal}(E/F)$ via two maps that are inverse to each other. And if you decode the writing there, you will get that $\text{Gal}(E/F)$ has fixed field $F$. If you want to look into more details, look at the part where they prove the two maps are inverse of each other.