Let $E$ be an elliptic curve over $\mathbb{Q}$ with a p-torsion point $P$ lying in $E(\mathbb{Q})$, where $p$ is a prime number. Let $e(\ ,\ )$ denote the Weil pairing.
Choose a point $Q$ such that $P, Q$ consist of a basis of $E[p]$, then we have the exact sequence $0\rightarrow \text{ker}f \hookrightarrow E[p]\xrightarrow{f} \mu_p\rightarrow 0$, where $f$ is the map $S\rightarrow e(S,P)$, and it is easy to know $\text{ker}f=\{\mathbb{Z}P\}$.
Let $K=\mathbb{Q}(E[p])$, so $Gal(K/\mathbb{Q})$ acts on $E[p]$ by $g(P)=P, g(Q)=xP+yQ$ for two integers $x, y$ and $p\nmid y$.
$\textbf{So the question is that}$ how to prove the short exact sequence metioned above splits if and only if $x=0$.
This is very important for me to prove Mazur's theorem which says that the torsion is not more than 13.
Thanks !