In [Deux Théorèmes d'Arithmétique](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-3/issue-1/Deux-Th%C3%A9or%C3%A8mes-dArithm%C3%A9tique/10.2969/jmsj/00310036.full) Chevalley shows the following related statement (Remarque p. 39): let $p$ be a prime, $K$ a field of characteristic different from $p$, and $\zeta$ a $p^e$-th primitive root of $1$ in some algebraic extension of $K$, where $e\geq 1$ is any integer. Assume moreover that $-1$ is a square in $K$ if $p=2$. Then if an element $x\in K(\zeta)$ is a $p^e$-th power, then it is already a $p^e$-th power in $K$.