= Solution
Classical <Kummer theory> relates extraction of $n$th roots to <Galois cohomology>. In characteristic zero the exact sequence
$$
1\longrightarrow\mu_n\longrightarrow\overline K^{\times}
\xrightarrow{z\mapsto z^n}\overline K^{\times}\longrightarrow1
$$
and <Hilbert theorem 90> identify $H^1(K,\mu_n)$ with $K^{\times}/K^{\times n}$. For an <elliptic curve>, the corresponding <Kummer exact sequence of an elliptic curve> is
$$
0\longrightarrow E[n]\longrightarrow E(\overline K)
\xrightarrow{[n]}E(\overline K)\longrightarrow0.
$$
Multiplication by $n$ is surjective over the <algebraic closure>. If $nQ=P\in E(K)$, the cocycle $\sigma\mapsto\sigma Q-Q$ takes values in $E[n]$. Changing the choice of $Q$ changes it by a coboundary, and changing $P$ by an element of $nE(K)$ does not change its class. Conversely, a trivial cocycle class allows $Q$ to be adjusted by an $n$-torsion point to become $K$-rational. Thus the <Kummer map of an elliptic curve> is an injection
$$
\delta:E(K)/nE(K)\hookrightarrow H^1(K,E[n]).
$$
This is the elliptic form of <Kummer theory>. The full cohomology group need not be finite; the arithmetic restriction on these classes is essential.
Choose a finite set $S$ of places containing the archimedean places, the <primes> over $n$, and all <primes> of <bad reduction of an elliptic curve>. At a finite place outside $S$, $E$ has <good reduction> and $n$ is a <unit>. A division point of the reduction of $P$ exists over the algebraic closure of the <residue field>. Smooth lifting gives a point over a finite <unramified extension> whose multiple differs from $P$ by an element of the <kernel of reduction of an elliptic curve>. In the <formal group of an elliptic curve>, $[n](T)=nT+O(T^2)$ is an isomorphism by the <invertible morphism criterion for formal group laws>, and its integral inverse converges on the maximal ideal. Correcting that difference produces a division point in the maximal <unramified extension>. Hence the <Kummer map of an elliptic curve> class is unramified outside $S$.
To prove finiteness explicitly, choose a finite <Galois extension> $L/K$ containing all $E[n]$ and all $n$th roots of unity, and enlarge $S$ to include its ramified places. A basis of $E[n]\cong(\mathbb Z/n\mathbb Z)^2$ identifies it over $L$ with $\mu_n^2$. Classical <Kummer theory> then identifies
$$
H^1(L,E[n])\cong(L^{\times}/L^{\times n})^2.
$$
The restriction of every class in the image of $\delta$ belongs to $L(S_L,n)^2$, where the <S-unramified power class group> is
$$
L(S_L,n)=\{[a]:v_{\mathfrak p}(a)\equiv0\pmod n\text{ for all }\mathfrak p\notin S_L\}.
$$
Indeed an unramified local <Kummer extension> at residue characteristic prime to $n$ has valuation divisible by $n$: in an unramified field containing a root, $n v(a^{1/n})=v(a)$ with integral valuations.
The <finiteness of S-unramified Kummer classes> follows from the exact sequence
$$
0\longrightarrow\mathcal O_{L,S_L}^{\times}/(\mathcal O_{L,S_L}^{\times})^n
\longrightarrow L(S_L,n)
\longrightarrow\operatorname{Cl}(\mathcal O_{L,S_L})[n]
\longrightarrow0.
$$
To see the final map, write the ideal of $a$ away from $S_L$ as $\mathfrak a^n$ and take the <ideal class> of $\mathfrak a$. Its kernel is represented by an <S-unit>, after division by an $n$th power; conversely an $n$-torsion <ideal class> yields such an $a$. The <S-unit group> is finitely generated by the <Dirichlet unit theorem> together with the finitely many inverted <primes>. The <ideal class group> of the localized ring is a quotient of the finite ordinary <ideal class group>. Both outer groups are therefore finite.
Finally restriction has finite kernel: inflation-restriction puts it in the finite group $H^1(\operatorname{Gal}(L/K),E[n])$. Thus the image of $\delta$ has finite restriction image and finite kernel, and
$$
\boxed{\#(E(K)/nE(K))<\infty\qquad(n\geq2).}
$$
\b[This proves the weak Mordell-Weil theorem.] Combining it with the <height descent lemma> proves the full <Mordell-Weil theorem>. Local restrictions at every place refine the finite group used here to the <Selmer group of an elliptic curve>, which is useful for explicit descent calculations.
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