As the title states, the setup is
Let $E = (\mathbb{Z}/(p))(t)$, and we are looking at it over $\mathbb{Z}/(p)$ and $t$ is transcendental.
Let $G$ be a group of automorphisms of $E$ generated by $\sigma: t \to t+1$. Determine $F = Inv \, G$, and $[E:F]$.
Adjoining a transcendental element to a finite field is one of my weakest aspects of Galois theory, so I don't really know where to start. So $\sigma^p = id$, so $|G| = p$?
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 $\sigma(P(t))=P(t)$?