Find the value of [K:$\mathbb{F}_{p}$] here?

Question

This question was asked in masters entrance and I was unable to solve it .

Let p be a prime number and F be a field of $p^{23}$ elements . Let $\phi$: F->F be the field automorphisms of F sending $a $ to $a^{p}$ .Let K={a belonging to F | $\phi(a) =a $}. What is the value of [K:$\mathbb{F}_{p}$]?

Attempt : The problem arised because I am not sure which elements of F will go to a under $\phi$ and how to compute those elements .

Answer 1

$K$ is simply $\Bbb F_p$. To prove this, you can argue as follows.

$\Bbb F_p \subseteq K$ because of Fermat's little theorem.

On the other hand $K$ is by definition the set of roots of the polynomial $x^p-x \in F[x]$. Since $F$ is a field, this polynomial has at most $p$ roots. This means that $K$ is a set with at most $p$ elements.

Thus $K= \Bbb F_p$.

Answer 2

The elements of $K$ are exactly the roots of the polynomial $x^p-x\in F[x]$ which belong to $F$. On the one hand we know that all the elements of $\mathbb{F_p}$ are such roots, this is Fermat's little theorem. On the other hand a polynomial of degree $p$ over a field can't have more than $p$ roots and hence these are all the roots, i.e $K=\mathbb{F_p}$. So the degree is $1$.