Let $F$ be a field and $E$ be a spliting field of some $f(x)\in F\left[x\right]$, how does one prove that
$$|\operatorname{Aut}(E/F)|\leq\left[E:F\right]?$$
I have no idea since I know $\operatorname{Aut}(E)$ can be humongous.
When does the equality hold?
In this context I believe $Aut(E/F)$ refers to the Galois group of $E/F$, which is the set of all $F$-automorphisms of $E$, as opposed to the group of ALL automorphisms of E. Since E is a splitting field it is finitely generated and it turns out such an automorphism is completely determined by its action on the generators. Equality holds when the extension is also separable. This is part of the statement of the Fundamental Theorem of Galois Theory. The proof is readily found in most Algebra texts. I can recommend Algebra by Mark Sepanski for its development of this particular topic.