$E$ is elliptic curve over field $k$. $K$ is the field extension of $k$ adjoining all $N$ torsion points. $K/k$ separable?

Question

Let $E$ be an elliptic curve over field $k$. $K$ is the finite field extension of $k$ s.t. all $N$ torsion points of $E$ are in $K$ coordinates. It is clear that $K/k$ is normal extension due to action of $Gal(\bar{k}/k)$ permuting $N$ torsion points solutions.

"$K/k$ is galois if $char(k)$ does not divide $N$"

$\textbf{Q:}$ How is $char(k)$ not dividing $N$ implying separability?

Ref. Lang, Elliptic Functions Chpt 2, Sec 1's field of N-division points.