Let $F$ be a finite field, and $h: F \to F$ be the map $f(x)=x^n$.
Find necessary and sufficient conditions for $h$ to be invertible and find the inverse of f in the case when it is invertible.
I am studying finite fields, I know that if $|F|=p^m$ for some integer $m$, then the extension $F/F_p$ is a Galois extension with a cyclic Galois group generated by the Frobenius map $x \mapsto x^p$.
So I guess the necessary and sufficient condition is that $n$ is the prime number $p$ with $|F|=p^m$. Is my guess correct? Thank you!
Well, $x\mapsto x^n$ is certainly a group homomorphism on the group of units $F^\times$. It is invertible if and only if it is one-to-one (since $F$ is finite), if and only if it has trivial kernel.
Note that $F^\times$ is cyclic of order $p^m-1$, in which case the power map $x\mapsto x^n$ acts on $F^\times$ the same way the multiplication map $u\mapsto nu$ acts on $\mathbb{Z}/(p^m-1)\mathbb{Z}$. This is invertible if and only if $n$ is itself invertible mod $p^m-1$, which is equivalent to $\gcd(n,p^m-1)=1$.