Weil pairing, Kummer theory, help to decrypt what  Wikipedia says I do not quite understand the sentence in the Wikipedia article:
http://en.wikipedia.org/wiki/Weil_pairing
Section "Formulation" line 3:
"... for given points $P,Q \in E(K)[n]$, where $E(K)[n]=\{T \in E(K) \mid n \cdot T = O \}$ and $\mu_n = \{x\in K \mid x^n =1 \} $, by means of [[Kummer theory]]. "
The sentence seems to me un-understandable. Can one comment ?
More mathematical question:
 what Kummer theory has to do with Weil pairing ?
 A: Let $G$ be a commutative algebraic group defined over a (number) field $K$. Write $G[n]$ for the kernel of the map $G(\overline{K})\to G(\overline{K})$ defined by $x\mapsto nx$. Let $K_n=K(G[n])$ be the field generated by $G[n]$. In this setting, Kummer theory is the theory of the extensions $K_n(y)$, where $y\in G(\overline{K})$ is a point satisfying $ny\in G(K)$. In one of the definitions of the Weil pairing on an elliptic curve, one pulls back the point $P\in E[n]$ by the multiplication-by-$n$ map and uses the resulting divisor to create a function, which is then evaluated (more or less) at the other $n$-torsion point. This procedure may be viewed as occurring in the Kummer extension $K([n]^{-1}(P),E[n])$. As Charles said, the heuristic remark in the Wikipedia may be helpful to some people, although maybe there's a way to phrase it so as to make it more helpful to more people. 
A: It is perhaps better to say that $\mu_n(\overline{K})$ admits a Galois action described by Kummer theory, and that the Weil pairing is Galois equivariant.  This does not require the assumption in the article that $K$ contains $n$th roots of unity.
A: The reason Kummer theory is involved is that the Galois covering of an elliptic curve E created by multiplication by n, assuming that n is prime to the characteristic of the base field K, has Galois group that is the product of two copies of a cyclic group of order n. Under the same assumption on the characteristic, when K is algebraically closed, there are n roots of unity of order n in K. Therefore when you look at the extension of function fields corresponding to the isogeny "multiply by n on E", Kummer theory can be applied. We know the Galois group is built up from two cyclic groups, so the extension of function fields is built up, by extracting two n-th roots of functions. That is what the remark says. It shouldn't be too hard to work out the details: the typical proofs for the pairing actually construct functions and translate them.
