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Kummer theory begins with the exact sequenceGalois cohomology and Hilbert theorem 90 identifyso cyclic extensions of exponent dividing $n$ are described by adjoining $n$th roots when $K$ contains ${\mu }_n$.

For an elliptic curve $E/K$, multiplication by $n$ gives the Kummer exact sequence of an elliptic curveIts connecting homomorphism is the injective Kummer map of an elliptic curvewhere $nQ=P$. Passing to the finite division field of an elliptic curve $K(E[n])$ makes $E[n]$ constant. Rational functions whose divisors are $n(T)-n(O)$ then express the classes through finitely many elements of $K(E[n]{)}^{\times }/K(E[n]{)}^{\times n}$.

Let $S$ contain the primes above $n$, the primes of bad reduction of an elliptic curve and the finitely many primes introduced by these functions. The local theory of good reduction shows that every Kummer class coming from $E(K)$ is unramified outside $S$, so it lies in an S-unramified power class group. Such a group is finite: valuations outside $S$ vanish modulo $n$, the ideal class group is finite, and the Dirichlet unit theorem makes the group of $S$-units modulo $n$th powers finite. The kernel created by passing to $K(E[n])$ is finite by finite-group Galois cohomology. HenceThis is the Kummer-theoretic proof of the weak Mordell-Weil theorem. Combining it with the height descent lemma proves the Mordell-Weil theorem: $E(K)$ is a finitely generated abelian group.

For a discrete module $M$ over the absolute Galois group $G_K$, the first Galois cohomology group iswhere a one-cocycle satisfies $c(\sigma \tau )=c(\sigma )+\sigma c(\tau )$ and a coboundary has the form $c(\sigma )=\sigma m-m$.

For an integer $n\ge 2$, the Kummer exact sequence of an elliptic curveproduces the injective Kummer map of an elliptic curveExplicitly, if $nQ=P$, then $\delta (P)$ is represented by $\sigma \mapsto \sigma Q-Q$.

For every completion $K_v$ there is a local Kummer map. The n-Selmer group isIt fits intowhere $\mathrm{Sha}(E/K)$ is the Tate–Shafarevich group.

Only finitely many places divide $n$, are places of bad reduction, or are Archimedean. Outside this finite set $S$, every Selmer class is unramified. Since the finite Galois module $E[n]$ has finite order, there are only finitely many $E[n]$-valued cohomology classes unramified outside $S$; equivalently, the relevant finite extensions have bounded degree and ramification, and their number is finite by the Hermite–Minkowski theorem. Hence ${\mathrm{Sel}}^{(n)}(E/K)$ is finite, and its subgroup $E(K)/nE(K)$ is finite. This is the Weak Mordell-Weil theorem. Combined with height descent, which chooses representatives of bounded height in the finitely many cosets modulo $nE(K)$, it yields the finite generation asserted by the Mordell-Weil theorem.

For a split CM prime $\mathfrak{p}$, use $E[{\mathfrak{p}}^{\infty }]$ in the Galois cohomology definition of the Selmer group of an elliptic curve, with local conditions given by its Kummer maps. The decomposition of the CM $p$-power torsion splits the full $p$-primary Selmer group into the two prime-primary pieces. Class-field descriptions must retain the CM-character twist and distinguish relaxed local conditions from the actual Selmer conditions.

A Sylow pro-$p$ subgroup of a profinite group is a closed pro-$p$ subgroup whose images are Sylow $p$-subgroups in every finite quotient. Such subgroups exist and are conjugate. Every open subgroup containing one has index prime to $p$. They allow prime-to-$p$ finite extensions to detect $p$-primary Galois cohomology.

For an abelian variety $A$ over a field $K$, its Weil–Châtelet group is the Galois cohomology group $H^1(K,A(K^{\mathrm{sep}}))$, classifying torsors under $A$. Such an algebraic-group torsor becomes $A$ after passing to a separable closure but need not have a $K$-point. A chosen geometric point identifies the torsor with $A$ and its Galois descent with a cocycle; changing that point adds a coboundary. The zero class is precisely the torsor with a rational point. For an elliptic curve, these torsors are genus-one curves with a specified action and Jacobian. Over a number field the kernel of restriction to every completion is the Tate–Shafarevich group.