extending rational functions over finite fields

Question

This is probably similar to the question in the link, but i'm not sure how to solve it either..

I want to prove $\mathbb F_p(t)/\mathbb F_p(t^p-t)$ is Galois, compute its Galois group, and describe the automorphisms. The extensions is Galois because I think finite fields are Galois over all their subfields and the former field is finite. Again i know i gotta use $t^p-t\equiv 0$ somehow, but i just don't see it :(

Proving an extension is galois and describe its automorphisms

Answer 1

Let $L=\Bbb{F}_p(t)$ and $K=\Bbb{F}_p(t^p-t)$, where $t$ is an indeterminate (that is, a transcendental element over $\Bbb{F}_p$). Because $t^p-t\in L$, we see that $L$ is an extension field of $K$. Below I split your task into small parts.

1. Consider the polynomial $$f(x)=x^p-x-(t^p-t).$$ Show that $f(x)\in K[x]$ and show that $f(t)=0$.
2. Show that for all $j=0,1,2,3,\ldots,p-1$, we also have that $$f(t+j)=0.$$ Hint: Here you need Little Fermat and what you know about expanding $(a+b)^p$ in a field of characteristic $p$.
3. Ok, so no we know $p$ zeros of $f(x)$ in $L$. Show that there aren't any others.
4. Show that $L$ is the splitting field of $f(x)$ over $K$.
5. Show that $f(x)$ is separable over $K$, and that $L/K$ is a Galois extension.
6. Show that $f(x)$ is the minimal polynomial of $t$ over $K$. Irreducibility of $f(x)$ may be a bit tricky, but has been explained for example here (sorry about blowing my own trumpet - there are several other good answers there, but some of them depend on facts that haven't been proven within this exercise yet). If you want to skip this, you need to show some other way that $[L:K]=p$.
7. Any $K$-automorphism $\sigma$ of $L$ must map $t$ to another zero of $f(x)$. What alternatives are there?
8. Identify the Galois group (I'm not sure whether this was asked, but you should do it anyway).