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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?

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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.

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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.

When you are done, see here for further discussion about the Frobenius map and its bijectivity (or lack thereof).

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