Let $K=\mathbb{F}_{p}(x)$, the function field in one variable, and let $G=\mathbb{F}_{p}$. The group $G$ acts as automorphisms of $K$ by $\sigma_{a}(x)=x+a$. Show that $K^{G}=\mathbb{F}_{p}(y)$, where $y=x^{p}-x$.

[State clearly any theorems you use.]

Is $K / K^{G}$ a separable extension?

Now let

$$H=\left\{\left(\begin{array}{ll} d & a \\ 0 & 1 \end{array}\right): a \in \mathbb{F}_{p}, d \in \mathbb{F}_{p}^{*}\right\}$$

and let $H$ act on $K$ by $\left(\begin{array}{ll}d & a \\ 0 & 1\end{array}\right) x=d x+a$. (The group structure on $H$ is given by matrix multiplication.) Compute $K^{H}$. Describe your answer in the form $\mathbb{F}_{p}(z)$ for an explicit $z \in K$.

Is $K^{G} / K^{H}$ a Galois extension? Find the minimum polynomial for $y$ over the field $K^{H}$.