= Solution
<Kummer theory> studies extracting roots through Galois actions, and on an <elliptic curve> the corresponding operation is division of points. The bridge is a cocycle: different Galois conjugates of a chosen division point differ by torsion. Controlling the ramification of these cocycles is what proves the <Weak Mordell-Weil theorem>.
Classically, for a characteristic-zero field $K$, the sequence
$$
1\longrightarrow\mu_m\longrightarrow\overline K^{\times}\xrightarrow{\ z\mapsto z^m\ }\overline K^{\times}\longrightarrow1
$$
is exact. <Hilbert's theorem 90> gives $H^1(K,\overline K^{\times})=0$, so <Galois cohomology> identifies
$$
H^1(K,\mu_m)\cong K^{\times}/(K^{\times})^m.
$$
The class of $a$ is represented by $\sigma\mapsto\sigma(\sqrt[m]{a})/\sqrt[m]{a}$. If $\mu_m\subset K$, the module is constant and these are continuous characters into $\mu_m$; the associated fields are cyclic Kummer extensions of degree dividing $m$. Several such classes give abelian extensions of exponent dividing $m$. Without the roots-of-unity hypothesis the cohomological description still holds, but the action on $\mu_m$ must not be discarded.
For an <elliptic curve> over a <number field> $K$, multiplication by $m\ge2$ gives the <exact sequence> of Galois modules
$$
0\longrightarrow E[m]\longrightarrow E(\overline K)\xrightarrow{[m]}E(\overline K)\longrightarrow0.
$$
The map is onto because it is a nonconstant isogeny over an algebraically closed field. For $P\in E(K)$ choose $Q\in E(\overline K)$ with $mQ=P$, and set
$$
\delta_m(P)_\sigma=\sigma Q-Q\in E[m].
$$
It satisfies $\delta(P)_{\sigma\tau}=\delta(P)_\sigma+\sigma\delta(P)_\tau$, the <Galois 1-cocycle> identity. Changing $Q$ by a <torsion point of an elliptic curve> changes the cocycle by a coboundary. If the class is zero, a corresponding change of $Q$ makes it Galois-fixed, so $P\in mE(K)$. Conversely a rational division point gives the zero class. Thus the <Kummer map of an elliptic curve> is injective on the quotient, and the cohomology sequence gives
$$
\boxed{0\longrightarrow E(K)/mE(K)\xrightarrow{\delta_m}H^1(K,E[m])\longrightarrow H^1(K,E)[m]\longrightarrow0.}
$$
This is the <Kummer exact sequence of an elliptic curve>. The last group parametrizes the appropriate torsor classes. It is important that the entire middle group need not be finite; an unrestricted Kummer class can ramify at arbitrarily many primes.
The <Weak Mordell-Weil theorem> states \b[$E(K)/mE(K)$ is finite for every <number field> $K$ and every $m\ge1$]. The case $m=1$ is trivial. To prove the remaining cases, choose a finite set $S$ containing the primes of bad reduction and the primes dividing $m$. At a prime outside $S$, the <elliptic curve> extends to a smooth proper <group scheme> over the valuation ring, and $[m]$ is finite étale. Properness extends a local <rational point> to a section; its division fibre is a finite étale torsor over that ring. Therefore its Kummer class is unramified. This is <unramified division torsors at good primes>, and shows that $\delta_m(E(K))$ lies in <first Galois cohomology unramified outside a finite set>.
Let $L=K(E[m])$, enlarged if necessary to contain $\mu_m$, and include in $S$ the finitely many primes ramified in $L/K$. This is a finite Galois extension. Over $L$ the torsion module is constant, isomorphic to $(\mathbb Z/m\mathbb Z)^2$, and so to two copies of $\mu_m$. Restricting a cocycle to $G_L$ therefore reduces it to two classical Kummer classes unramified outside the primes above $S$. Such a class is represented by $a\in L^{\times}$ satisfying
$$
v_{\mathfrak p}(a)\equiv0\pmod m\qquad(\mathfrak p\notin S_L).
$$
Write this power-class group as $L(S_L,m)$. The implication follows locally because, away from residue characteristic dividing $m$, the valuation of an unramified $m$th-root extension must be divisible by $m$.
Its finiteness comes from two standard arithmetic finiteness results, rather than from an assertion that all of $L^{\times}/(L^{\times})^m$ is finite. The valuations outside $S_L$ associate to $a$ an ideal whose $m$th power is principal. There is an <exact sequence>
$$
0\longrightarrow\mathcal O_{L,S_L}^{\times}/(\mathcal O_{L,S_L}^{\times})^m\longrightarrow L(S_L,m)\longrightarrow\operatorname{Cl}(\mathcal O_{L,S_L})[m]\longrightarrow0.
$$
The left group is finite by the finite generation of the <S-unit group>, and the right group is finite by <finiteness of the ideal class group>. This proves <finiteness of S-unramified Kummer classes>. Hence there are only finitely many restricted cocycles over $L$. The kernel of restriction from $K$ is contained in $H^1(\operatorname{Gal}(L/K),E[m])$, which is finite because both group and module are finite. The restricted cohomology group over $K$ is therefore finite, and its injected subgroup $E(K)/mE(K)$ is finite too. This completes a <Kummer-theoretic proof of the weak Mordell-Weil theorem>.
For arithmetic computation one adds local conditions. The <Selmer group of an elliptic curve> consists of classes in $H^1(K,E[m])$ whose restriction at every completion $K_v$ lies in the image of the corresponding local Kummer map. They are unramified outside the same finite set, so this group is finite. The failure of a everywhere locally soluble torsor to have a global point is measured by the <Tate–Shafarevich group>, defined as the kernel of the local restriction map on $H^1(K,E)$. The resulting <exact sequence> is
$$
0\longrightarrow E(K)/mE(K)\longrightarrow\operatorname{Sel}^{(m)}(E/K)\longrightarrow\operatorname{Sha}(E/K)[m]\longrightarrow0.
$$
Thus the <Selmer group of an elliptic curve> gives a computable upper bound for the quotient; actual <rational points> supply lower bounds. Equality need not follow just from local solubility. Finiteness of this fixed torsion subgroup does not prove finiteness of the whole <Tate–Shafarevich group>.
Finally, weak finiteness alone is not finite generation: the additive group $\mathbb Q$ has trivial quotients by multiplication and is not finitely generated. The additional ingredient for the <Mordell-Weil theorem> is <height descent>. Choose representatives $P_i$ of $E(K)/mE(K)$ and write $P=P_i+mQ$. The quadratic growth of the <canonical height of an elliptic curve>, together with bounds for translation by the finitely many $P_i$, makes $Q$ have smaller height whenever $P$ is sufficiently large. <Northcott theorem> gives only finitely many points of bounded height, and repeated descent yields a finite generating set. This explains the roles of <Kummer theory>, local information and heights without conflating the weak and full theorems.
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