Preimage of multiplication n-isogeny on a point of elliptic curve generates a Galois extension

Question

Let $E$ be an elliptic curve over a number field $K$ such that $E(K)$ contains all $n$-torsion points of $E$. Why is $K([n]^{-1}P)/K$ a Galois extension, for all points $P$? If the hypothesis that about $n$-torsion points is removed, is this still true?

Answer 1

Your teacher meant that if $E(K)$ contains $E[n]$ and $Q$ is a point such that $[n]Q\in E(K)$ then $K(Q)/K$ is a Galois extension.

If $E(K)$ doesn't contain $E[n]$ and $S$ is the set of points $\{ Q\in E, [n]Q=P\}$ with $P\in E(K)$ then $K(S)/K$ is a Galois extension.

For the notation $[n]^{-1}P$: it is a well-defined element of the quotient group $E/E[n]$, so a set of elements of $E$.