# 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 ?

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@Scott Ooops... Thank You !!! –  Alexander Chervov Jan 25 '12 at 12:45

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.

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@Joe thank You for the answer. May I ask for the clarification of the following issue - the "relation Kummer<-> Weil" is just on the level that some objects are involved are similar or there is deeper relation (I guess it is) ? I would expect that class field theory for Kummer extensions has something to do with Weil pairing - I mean that probably Weil pairing is related to some symbol (tame, Hilbert???) is there something like that ? PS I like heuristics very much, just I like when it is clearly written - "this is heuristics, comes as it is"... –  Alexander Chervov Jan 26 '12 at 8:07

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.

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@Scott Thank You again ! That sounds quite clarifying and reasonable - any natural construction like Weil pairing should be Galois equivariant. And it seems from what is written below the Weil pairing is so... –  Alexander Chervov Jan 25 '12 at 12:53
Galois equivariance - it seems we do not need Kummer theory - it is due to everything is defined algebraically, is it correct ? So not quite clear why Kummer has to do with this. –  Alexander Chervov Jan 25 '12 at 13:08