# Frobenius Auto need not be an automorphism if F is infinite

I'm trying to find an example to show the map $\sigma_p : F \rightarrow F$ given by $\sigma_p(a)=a^p$ for $a\in F$ need not be an automorphism in the case that F is infinite. I'm lost as to where to start. Any pointers?

Simply pick any field where not all elements are $p$th powers.
For example, let $F=\mathbb F_P(t)$, the field of rational functions on the variabe $t$ with coefficients in the prime field $\mathbb F_p$. You should have no problem showing that $t\in F$ is not a $p$th power.
Here is a discussion of two standard examples of infinite fields of characteristic $p$. One of them is a counterexample, and the other is not. Try to work out for yourself which is which.