Let $\mu \in \mathbb{Q}(\zeta_n)$ lie above the rational prime $p$, and let the prime ideal $\mathscr{P}\subset \mathbb{Z}[\zeta_n]$ have ramification index $a$ over $\mu$.

Why is it then true that $\mathscr{P}$ has ramification index $\frac{n}{(a,n)}$ over $p$?

In "Reciprocity Laws: from Euler to Eisenstein'', it's stated that this is from the decomposition law in Kummer extensions (as $\mathbb{Q}(\zeta_n, \mu^{1/n})/\mathbb{Q}(\zeta_n)$ is Kummer) but I cannot find any reference about this. Does anyone know of one?

I assume it's irrelevant, but just in case: $\mu$ here is the $n$th power of a Gauss sum of a residue character.