How to prove $t(x)-x\in F_{p}$ when $p(x) \in K$, $K$ a field whose characteristic is $p>0$

Question

Suppose $K$ is a field whose characteristic is $p>0$, and $L$ is a finite Galois extension of $K$. Also, we have an endormorphism $h$ such that $h(x)=x^p-x$ for any $x \in L$.

Question: If $t\in \operatorname{Gal}(L/K)$ and $x\in L$, and we also have $h(x)\in K$,then we can prove that $t(x)-x$ is contained in $F_{p}$

This question is from my algebra lecture, but I still couldn't prove it, can someone tell me how to prove it?

I think I need to prove $(t(x)-x)^p=t(x)-x$,but I still cannot prove it Also, $(t(x)-x)^p=(t(x))^p+(-x)^p$, how can I get $t(x)-x$?

Answer 1

Here's a plan:

1. Show that in $\Bbb{F}_p[T]$ we have the factorization $$h(T):=T^p-T=\prod_{i=0}^{p-1}(T-i). $$
2. Recall that $h(x+y)=h(x)+h(y)$ for all $x,y\in L$.
3. So if $a=h(x)\in K$, show that the minimal polynomial of $x$ over $K$ is a factor of $$r(T)=T^p-T-a\in K[T].$$
4. Show that the zeros of $r(T)$ are $x+i,i\in\Bbb{F}_p$.
5. Prove the claim.