Let $k\subset K$ be an extension having degree $[K:k]=n$ coprime to $p$. Prove that $a$ is a $p$th power in $k$ if only if it is in $K$
This is a problem in Galois theory - Miles Ried. I'm learning Galois theory by myself. I can't understand $p$th power, i think is that $a^p$?
If $p=2$ and $a$ is not a square in $k$, then the polynomial $X^2-a $ is irreducible over $k$ and if it had a root $\alpha \in K$ the tower $k\subset k(\alpha)\subset K$ would imply that $n$ is even, a contradiction.
If $p$ is odd and $a$ is not a $p$-th power in $k$, then $f(X)=X^p-a$ is irreducible and again we conclude that since $p$ is coprime to $n$ no root $\alpha$ of $f(X)$ can be contained in $K$.
That irreducibility of $f(X)$ is however a rather difficult theorem due to Capelli and proved in Lang's Algebra: Chapter VI, theorem 9.1, page 297 of the Third Edition.