Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion points of $E$. Then if $m=p_1^{r_1}\cdots p_k^{r_k}$ where $p_i$, $i=1,2,\cdots,k$ are prime numbers, can we say that $Gal(K(E[m])/K)\cong Gal(K(E[p_1^{r_1}])/K)\times\cdots\times Gal(K(E[p_k^{r_k}])/K)$?
If yes, then is it easier to directly prove the isomorphism above or the isomorphism $Gal(K(E[p_1^{r_1}])\cdots K(E[p_k^{r_k}])/K)\cong Gal(K(E[p_1^{r_1}])/K)\times\cdots\times Gal(K(E[p_k^{r_k}])/K)$?
I understand that the second isomorphism will only hold if $K(E[p_i^{r_i}])\cap K(E[p_j^{r_j}])=K$, where $1\leq i,j\leq k$, $i\neq j$. But I am unable to proceed after this. If somebody could help me or give me a hint I would really appreciate it.
It is certainly true that $K(E[m])/K$ is the composite field of the $K(E[p_i]^{r_i})$, so your question boils down to the Galois theory of the composite fields (the Galois group of $KL/F$ is isomorphic to the Galois group of $K/F$ times the Galois group of $L/F$ if and only if $K \cap L = F$).