I am working on this problem :
Let $a>1$ be a square-free integer. For any prime number $p>1$, denote by $E_p$ the splitting field of $X^p-a \in Q[X]$ and for any integer $m>1$, let $E_m$ be the composition of all $E_p$ for all primes $p|m$. Compute the degree $[E_m:\mathbb Q]$
My thought is $E_p = \mathbb Q(\xi_p,a^{\frac{1}{p}})$, where $\xi_p = e^{\frac{2\pi i}{p}}$. Assume that all of the prime factors of $m$ are $p_1,\cdots,p_n$, then $E_m$ is the composite of $E_{p_1},\cdots,E_{p_n}$, which is $\mathbb Q(\xi_{p_1},a^{\frac{1}{p_1}},\cdots,\xi_{p_n},a^{\frac{1}{p_n}})$. Am I correct now?
If $n = 1$, the answer is $p(p-1)$, but I have no idea of general $n$. Thanks for help.
The splitting field of $X^p-a$ for $a$ squarefree is indeed $E_p=\Bbb Q(\zeta_p,\sqrt[p]{a})$, which has degree $p\varphi(p)=p(p-1)$ over $\Bbb Q$. For two such splitting fields $M$ and $N$ we have $$ [MN:\Bbb Q]=\frac{[M:\Bbb Q] [N:\Bbb Q]}{[M\cap N:\Bbb Q]}. $$ Can you take it from there?