Let $k$ be an algebraically closed field, let $n$ be an integer $(n, car.k) = 1$. Let $E$ be an elliptic curve $k$. Let $\{T_1, T_2\}$ be the base of $E[n] = \mathbb{Z}/n \oplus \mathbb{Z}/n$. Then how do I see that $e_n(T_1, T_2)$ is a primitive $n$th root of unity, where $e_n$ is the Weil pairing?
Update. I am using the following definition of Weil pairing.
Just to get this off the unanswered list--sorry for the delayed response, I didn't see your reply to my above comment.
Suppose that $e(T_1,T_2)$ were not primitive. Then, by definition, the exists some $m<n$ such that $e(T_1,T_2)^m=1$. But, note then that this says $e(mT_1,T_2)=1$. But, note then that for any $T\in E[n]$ we have that $T=\alpha T_1+\beta T_2$ for some $\alpha,\beta\in\mathbb{Z}$ we then have that
$$e(mT_1,T)= e(mT_1,T_1)^\alpha\cdot e(mT_1,T_2)^\beta=e(T_1,T_1)^{\alpha m}\cdot e(mT_1,T_2)^\beta=1^{\alpha m}\cdot 1^\beta=1$$
and thus, by the non-degeneracy, we have that $mT_1=0$. But, $T_1$ is part of a bsis of $E[n]$ and thus $|T_1|=n$. We arrive at a contradiction.