Field of rational functions over $\mathbb{F}_p$

Question

Let $K=\mathbb{Z}_p(x,y)$ be the field of rational functions of variables $x,y$ with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime.

Let $g(t)=t^p-x, h(t)=t^p-y \in K[t]$ and $E$ is the splitting field of the polynomial $f(t)=g(t) \cdot h(t) \in K[t]$.

Show that:

1. $g(t)$ and $h(t)$ are irreducible polynomials of $K[t]$.
2. $[E:K]=p^2$
3. The extension $K \leq E$ is not separable.
4. $a^p \in K$ for each $a \in E$

Could you give me some hints how to show the above??

Is the splitting field $E=K(x^{\frac{1}{p}}, y^{\frac{1}{p}})$ ??

Answer 1

Hints.

1. Use this for $n=1$.

2. Use this tower of fields: $K\subset K(\sqrt[p]{x})\subset E$.

3. $\sqrt[p]{x}$ is not separable over $K$.

4. Use the freshmen's dream.

Yes, $E$ is the splitting field of $f$.