Determine the fixed subfield [duplicate]
Solution 1:
As you have noticed, $\sigma^p=\operatorname{id}$, and clearly $\sigma$ is not identity, so $G$ has order $p$.
To calculate $[E:F]$ recall that the extension degree from the fixed field of the Galois group is always equal to the order of the group.
Finding $F$ shouldn't be too hard. Can you find a single nonconstant polynomial $P(t)$ such that $\sigma(P(t))=P(t)$?