# Full $n$-torsion of elliptic curves and the cyclotomic field of order $n$

Hi, overflowers.

I have a question concerning the torsion of elliptic curves over number fields.

Let us consider an elliptic curve $E$ defined over ${\mathbb Q}$. From the Weil pairing one can prove that, for all primes $p$, ${\mathbb Q}[\zeta_p] \subset {\mathbb Q}(E[p])$, where ${\mathbb Q}(E[p])$ is the algebraic extension generated by the coordinates of all points of order $p$ on $E$.

I was thinking whether this could also be true for all integers $n$, not necessarily prime. It is probably something well-known, but I can not find an appropriate reference, nor an easy counterexample.

Thanks a lot!!

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You can use the Weil pairing to show that same for all integers $n$. This is corollary 8.1.1 in Silverman's The Arithmetic of Elliptic Curves, in the section titled The Weil Pairing. –  Dror Speiser May 24 '13 at 8:34
Yes. Weil pairing applies to $n$-torsion too. –  i707107 May 24 '13 at 8:35
Thanks, I knew I was missing something! –  Chema Tornero May 24 '13 at 8:59