# Non-Galois number fields extension of prime degree

Let $E/F$ be an extension of number fields of degree $p$ prime which I know to be NON-Galois. By the fundamental theorem of Galois Theory, I can say that $|\operatorname{Aut}(E/F)|<p$, but can I say more about $|\operatorname{Aut}(E/F)|$?

Thank you in advance to whoever will help me.

Your set of automorphisms is a group, and its fixed field $k$ not only satisfies $F\subset k\subset E$, but also has $\vert\text{Aut}(E/F)|=[E:k]$. That’s the theorem that when you have a finite group of automorphisms of a field, the order of the group is the degree of the big field over the fixed field. Since $[E:k]=1$ necessarily, your automorphism group is trivial.