# extending rational functions over finite fields

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

• The fields you are talking about, e.g., $F_p(t)$, are not finite. – Gerry Myerson Jul 20 '16 at 12:36
• What is $t$? Usually I would assume it is a transcendental element but this does not match the fact that you said the field is finite. Also is it really $t^p -t$? – quid Jul 20 '16 at 12:36
• I'd say that $\mathbb F_p(t)$ is a extension of $\mathbb F_p(t^p-t)$, not the other way around. Maybe the notation is confused? – Thomas Andrews Jul 20 '16 at 12:46
• Sorry guys, my mistakes! the question was already edited though – user355165 Jul 20 '16 at 12:52
• @GerryMyerson right.. we can have arbitrarily large exponents. gotcha, sorry – user355165 Jul 20 '16 at 12:53

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).
• Fairly sure that this is a duplicate in the sense that all parts have been covered. Not sure if they all were done within a single thread. Will search later. Missus wants something, and today is our 25th anniversary, so I will obey :-) – Jyrki Lahtonen Jul 20 '16 at 14:55
• Really instructive answer! I'm trying to follow it myself but I'm stuck on the final two points. It seems the only automorphisms are translations, of which there are.. $(p-1)$ (?) How to "identify" the Galois group now? – user153312 Jul 20 '16 at 15:18
• @Exterior: Don't forget the trivial automorphism $t\mapsto t+0$. – Jyrki Lahtonen Jul 20 '16 at 15:23
• Okay, so $p$ automorphisms. How does this pinpoint the Galois group though? – user153312 Jul 20 '16 at 15:24
• You have your automorphisms. They compose nicely to form a group. They are defined on $\Bbb F_p(t)$, and their fixed field is $\Bbb F_p(t^p-t)$. Got the picture? – Lubin Jul 20 '16 at 15:33