If $K$ is an extension of $\mathbb{Z}/p\mathbb{Z}$, every elements of $K$ is a root of $t^{p^n -1}-1$

Question

Problem :

1. Show that the characteristic of a finite field $K$ is a prime number.

2. Show that $K$ is an extension of $\mathbb{Z}/p\mathbb{Z}$ and there exists $n$ such that Card($K$) $= p^n$.

3. Show that every non zero elements of $K$ is a root of $t^{p^n -1}-1$.

4. Show that $K$ is the splitting field of $t^{p^n}-t$.

I managed to prove the first and second problems. The first one by contradiction and the second by showing that $\mathbb{Z}/p\mathbb{Z}$ was isomorphic to a subfield of K (the image of the characteristic homomorphism).

However I have difficulties showing the third problem. I think it would equivalent to showing that $\forall x \in K : x^{p^n -1} = 1$, or maybe that $\forall x \in K : x^{p^n} = x$ and use Ferma's little theorem.

Answer 1

You're on the right track, but there are a couple of slips in your equivalences. The equation $\ x^{p^n-1} = 1\ $ doesn't hold for all $\ x\in K\ $, only for all non-zero $\ x\in K\ $, but that's precisely what it is that you're being asked to show in the third leg of your problem. And you can indeed use Lagrange's theorem (a generalisation of Fermat's little theorem) to show it, since the nonzero elements of $\ K\ $ form a multiplicative group of order $\ p^n-1\ $.

The other slip (just a typo, I presume) is in the equation $\ x^{p^n} = p\ $. This should be $\ x^{p^n} = x\ $.