Representation of $m$-th division field extension of an elliptic curve as composite field extensions

Question

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 $m=p_1^{r_1}\cdots p_k^{r_k}$, can we say that $K(E[m])/K\cong K(E[p_1^{r_1}])\cdots K(E[p_k^{r_k}])/K$, where $p_i$, $i=1,2,\cdots,k$ are prime numbers? And, on a related note, can we say that $Gal(K(E[m])/K)\cong Gal(K(E[p_1^{r_1}])\cdots K(E[p_k^{r_k}])/K)$?