Is $p(x^n)$ solvable by radicals if $p(x)$ is solvable by radicals?

Question

I have a question related to solvable by radicals polynomials. Is there any theorem or a result that assures me that if $p(x) \in \mathbb{Q}[x]$ is solvable by radicals then $p(x^n)$ is solvable by radicals for some $n \in \mathbb{N}$? If not, is there anything similar?

Answer 1

Your $p(x^n)$ is indeed always solvable, but I don't think this result has a particular name. Assume $p(x)$ is solvable by radicals. If you want a root of $p(x^n)$, just pick a root of $p(x)$, and take any of its $n$-th roots. This works for any positive integer $n$.