= Solution
Let $K$ be a field, $K_s$ a separable closure and $G_K$ its <absolute Galois group>. A discrete <Galois module> $A$ is an abelian group with a continuous $G_K$-action, so each element has an open stabilizer. <Galois cohomology> uses continuous cochains. Its zeroth group is $H^0(K,A)=A^{G_K}$, and its first group is
$$
H^1(K,A)=Z^1(K,A)/B^1(K,A),
$$
where
$$
c_{\sigma\tau}=c_\sigma+\sigma c_\tau,
\qquad
B^1(K,A)=\{c_\sigma=\sigma a-a:a\in A\}.
$$
Thus cocycles measure a failure of descent, and coboundaries account for changing a chosen lift. A short exact sequence of <Galois modules> gives a <long exact sequence in group cohomology>. Its connecting map sends an invariant element of a quotient to the cocycle $\sigma b-b$ of a chosen lift $b$; changing the lift changes this by a coboundary. These constructions provide the bridge between arithmetic divisibility and cohomology.
For <Kummer theory>, assume the characteristic does not divide $n$. The multiplicative exact sequence is
$$
1\longrightarrow\mu_n\longrightarrow K_s^\times
\xrightarrow{,n,}K_s^\times\longrightarrow1.
$$
The final arrow is surjective because a root of $T^n-a$ is separable for $a\ne0$. <Hilbert's theorem 90> says $H^1(K,K_s^\times)=0$. Here is its usual finite-extension proof. For a multiplicative cocycle $c_\sigma$ on a finite <Galois extension> $L/K$, choose $a\in L$ so that
$$
A=\sum_{\sigma\in\operatorname{Gal}(L/K)}c_\sigma\sigma(a)\ne0.
$$
Such an $a$ exists by the <Artin independence theorem>. The cocycle identity gives $\tau A=c_\tau^{-1}A$. With $b=A^{-1}$ this becomes $c_\tau=\tau b/b$, a coboundary. Every continuous cocycle with values in $K_s^\times$ is defined over a sufficiently large finite Galois extension, so the same result holds for the absolute group.
The <long exact sequence in group cohomology> now gives
$$
\boxed{H^1(K,\mu_n)\cong K^\times/(K^\times)^n.}
$$
The class of $a$ maps to $c_\sigma=\sigma(\alpha)/\alpha$, where $\alpha^n=a$. This cohomological statement does not require $\mu_n\subset K$; it retains the actual Galois action on $\mu_n$. If all $n$th roots of unity do lie in $K$, the action is trivial, and after choosing a primitive root of unity the cocycles are characters into $\mathbb Z/n\mathbb Z$. Thus adjoining an $n$th root of $a$ describes a cyclic extension of degree dividing $n$, and Kummer classes classify the corresponding characters. For $n=2$, this gives the familiar quadratic extensions and the <square-class group of a field>.
Now let $K$ be a <number field> and $E/K$ an <elliptic curve>. The <multiplication-by-n morphism> is a surjective isogeny on $E(\overline K)$, with kernel $E[n]\cong(\mathbb Z/n\mathbb Z)^2$. The <Kummer exact sequence of an elliptic curve>
$$
0\longrightarrow E[n]\longrightarrow E(\overline K)
\xrightarrow{[n]}E(\overline K)\longrightarrow0
$$
yields
$$
0\longrightarrow E(K)/nE(K)\xrightarrow{\delta}
H^1(K,E[n])\longrightarrow H^1(K,E)[n]\longrightarrow0.
$$
Explicitly, choose $Q$ with $nQ=P$ and set $\delta(P)_\sigma=\sigma Q-Q$. This lies in $E[n]$, satisfies the cocycle identity and is unchanged as a class if $Q$ is replaced by another division point. If it is a coboundary $\sigma T-T$ for $T\in E[n]$, then $Q-T$ is rational over $K$ and $P=n(Q-T)$. Conversely such a rational division point makes the cocycle zero. This proves the injectivity of the <Kummer map of an elliptic curve> on $E(K)/nE(K)$.
To prove the <Weak Mordell-Weil theorem>, it remains to prove that this particular cohomological image is finite. The full group $H^1(K,E[n])$ need not be finite, so the injection alone is not enough. Choose a finite set $S$ of finite places containing the bad-reduction places and those dividing $n$. At any other place, the construction in Question 3 works over the local field: reduce $P$, find an $n$-division point over a finite residue extension, lift it by the <Hensel lemma>, and correct its error by the inverse formal multiplication. Thus $P$ has an $n$-division point over the <maximal unramified extension>. Choosing that lift locally makes the cocycle zero on inertia. For an arbitrary lift its restriction is a coboundary, so the cohomology class is unramified. Therefore the image lies in <first Galois cohomology unramified outside a finite set>:
$$
\delta(E(K)/nE(K))\subseteq H^1_S(K,E[n]).
$$
We prove the required finiteness of this group. Take a finite Galois extension $L/K$ containing all of $E[n]$ and $\mu_n$, and enlarge $S$ to include any remaining ramification of $L/K$. Let $S_L$ be the places above it. The <inflation-restriction exact sequence> bounds the kernel of restriction to $L$ by
$$
H^1(\operatorname{Gal}(L/K),E[n]),
$$
a finite group, since both its group of arguments and its group of values are finite. Over $L$ the torsion module is constant and can be identified with $\mu_n^2$, so <Kummer theory> gives
$$
H^1(L,E[n])\cong\bigl(L^\times/(L^\times)^n\bigr)^2.
$$
Restrictions of our unramified classes remain unramified outside $S_L$. If a Kummer class $[a]$ is unramified at such a place $v$, its $n$th root can be taken in an unramified local extension. The extended valuation still has integer values, and $n\,v(\sqrt[n]a)=v(a)$. Hence $v(a)$ is divisible by $n$. Each coordinate therefore lies in the <S-unramified power class group>
$$
L(S_L,n)=\{[a]\in L^\times/(L^\times)^n:
v(a)\equiv0\pmod n\text{ for }v\notin S_L\}.
$$
This group is finite. Indeed, the ideal of $a$ away from $S_L$ is $\mathfrak a^n$. Sending $[a]$ to $[\mathfrak a]$ yields the exact sequence
$$
0\longrightarrow\mathcal O_{L,S_L}^{\times}/(\mathcal O_{L,S_L}^{\times})^n
\longrightarrow L(S_L,n)
\longrightarrow\operatorname{Cl}(\mathcal O_{L,S_L})[n]\longrightarrow0.
$$
For its kernel, a principal $\mathfrak a$ lets us divide $a$ by an $n$th power to obtain an $S_L$-unit. For surjectivity, an ideal class killed by $n$ has an $n$th power which is principal away from $S_L$, giving such an $a$. The <S-unit group> is finitely generated and the <ideal class group> is finite; hence both end groups are finite, proving <finiteness of S-unramified Kummer classes>. Restriction has finite image in $L(S_L,n)^2$ and finite kernel, so $H^1_S(K,E[n])$ is finite. We conclude
$$
\boxed{\#\bigl(E(K)/nE(K)\bigr)<\infty\qquad(n\ge2),}
$$
which is the <Weak Mordell-Weil theorem> and its <Kummer-theoretic proof of the weak Mordell-Weil theorem>.
This framework also defines the <Selmer group of an elliptic curve> by imposing that a class in $H^1(K,E[n])$ lie in the local Kummer image at every completion. Its classes are unramified outside a fixed finite set, so the same finiteness proof applies. Its relation to rational points is
$$
0\longrightarrow E(K)/nE(K)\longrightarrow
\operatorname{Sel}^{(n)}(E/K)\longrightarrow
\operatorname{Sha}(E/K)[n]\longrightarrow0.
$$
The <Tate–Shafarevich group> here measures the difference between locally soluble torsors and globally soluble ones. Finally, the weak theorem gives finite divisibility quotients; adding the <height descent lemma> and bounded-height finiteness proves the stronger <Mordell-Weil theorem>. Those height inputs are additional to the cohomological finiteness just established.
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