If $\sigma_{1}, \sigma_2, \ldots, \sigma_{n}$ is a group of automorphisms of a field $E$ and if $F$ is a fixed field of $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{n},$ then $(E/F) = n$.
If $\sigma_{1}, \sigma_2, \ldots, \sigma_{n}$ is a group of automorphisms of a field $E$ and if $F$ is a fixed field of $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{n},$ then $(E/F) = n$.
There is a result due to Artin . Let $G$ be a finite group of automorphism of $E$, let $F=E^G$ be the fixed subfield, then $E/F$ is finite and $[E:F] \leq |G|$. Note, you need the $\leq$ sign.