Editing Kummer theory (section)

== For Elliptic Curves ==
Kummer theory is often used in the context of elliptic curves. Let <math>E/K</math> be an elliptic curve. There is a short exact sequence

<math>0\xrightarrow{} E[m]\xrightarrow{} E \xrightarrow{P\mapsto m\cdot P} E \xrightarrow{} 0</math>,

where the multiplication by <math>m</math> map is surjective since <math>E</math> is divisible. Choosing an algebraic extension <math>L/K</math> and taking cohomology, we obtain the Kummer sequence for <math>E</math>:

<math>0\xrightarrow{} E(L)/mE(L)\xrightarrow{} H^1(L, E[m])\xrightarrow{} H^1(L,E)[m]\xrightarrow{}0</math>.

The computation of the weak Mordell-Weil group <math>E(L)/mE(L)</math> is a key part of the proof of the [[Mordell–Weil theorem|Mordell-Weil theorem]].  The failure of <math>H^1(L,E)</math> to vanish adds a key complexity to the theory.