Hi,


Suppose that $E/F$ is a Galois extension. If $P(X)\in E[X]$ is a (EDIT: monic) polynomial of
degree $n > 0$, such that $P(X)^n\in F[X]$, does it follow that $P(X)\in F[X]$?


Thanks



nth-powers and degree n polynomials with coefficients in field extensions
If $P$ is monic, the answer is Yes (in any characteristic, if the extension is separable), because a field automorphism over $F$ fixes $P^n$ and $(P^{\sigma})^n=P^n$ implies $P^{\sigma} = P$ for monic polynomials. In general, if $c$ is the leading coefficient of $P$, then $c^n \in F$ and
$c^{-1}P\in F[X]$. Thus the counterexamples from the comments are the only ones.


