Let $\mathbb{F}$ be a field of characteristic zero, and $\mathbb{F}[\alpha]$ its radical extension, i.e. $\alpha$ is a zero of $X^n-a \in \mathbb{F}[X],$ and $X^n-a$ is irreducible in $\mathbb{F}.$ Let $n=pq,$ where $p$ is a prime, and $q$ is a positive integer greater than $1$ (otherwise it would already be a prime radical extension. I'm trying to prove that every tower of radical extensions can be extended to a tower of prime radical extensions, where the degree of every extension is prime); therefore, $[\mathbb{F}[\alpha] : \mathbb{F}]=n.$
Then, $\alpha^q$ is a root of $X^p - a \in \mathbb{F}[X],$ so $\mathbb{F}[\alpha^q]$ is an extension of $\mathbb{F}.$ In order to show that it's also a radical extension, I'd need to prove that $X^p-a$ is irreducible in $\mathbb{F},$ i.e. that it has no zeroes in $\mathbb{F}.$ How can I do this? I've tried assuming the opposite and trying to get a contradiction, with no success.