Let $p$ be a prime and let $K$ be a field of characteristic $d$ co-prime to $p$. Let $a\in K$. I want to show, $a\in (K^*)^p\iff a\in (K(\zeta_p)^*)^p$, and this would eventually imply $\zeta_p\in K$.
If $a\in (K^*)^p$, as $(K^*)^p\subset (K(\zeta_p)^*)^p$, $a\in (K(\zeta_p)^*)^p$. If $a\in (K(\zeta_p)^*)^p$, $a=(a_0+a_1\zeta_p+...)^p$, therefore, taking norms we get $a^k=a_n^p$. Here, $k$ is $[K(\zeta_p):K]$, and $a_n=norm(a_0+a_1\zeta_p+...)$. As, $k$ is coprime to $p$, $kx+py=1$. Therefore, $a^{kx}=a^{1-py}\Rightarrow a=(a_n^xa^{y})^p$. As, $a_n^xa^{y}\in K$, we get that $a\in (K^*)^p$. Now, from this how to show $\zeta_p\in K$. One idea is to show there is another root of $x^p-a$ in $K$, but how to show that?