= Solution
\b[Galois cohomology and the finite division quotient.] Let $K$ be a <number field>, $G_K=\operatorname{Gal}(\overline K/K)$ with its profinite topology, and $M$ a discrete continuous $G_K$-module. Its <Galois cohomology> is continuous <group cohomology>. In degrees zero and one,
$$
H^0(K,M)=M^{G_K},\qquad
H^1(K,M)=\frac{\{c:G_K\to M\text{ continuous}:c(\sigma\tau)=c(\sigma)+\sigma c(\tau)\}}
{\{c(\sigma)=\sigma m-m:m\in M\}}.
$$
Thus a <crossed homomorphism> measures the failure of a choice to be Galois-invariant, and changing the choice changes the cocycle by a coboundary. A short exact sequence of such modules gives a <long exact sequence in group cohomology>. The connecting map sends an invariant element to the cocycle obtained by lifting it and comparing its Galois translates.
For an <elliptic curve> $E/K$ and $n\geq2$, multiplication by $n$ is surjective on $E(\overline K)$ and has kernel $E[n]$. The <Kummer exact sequence of an elliptic curve> therefore gives
$$
0\longrightarrow E(K)/nE(K)\xrightarrow{\kappa}H^1(K,E[n])
\longrightarrow H^1(K,E)[n]\longrightarrow0.
$$
Explicitly, choose $nQ=P$ and set $\kappa(P)(\sigma)=\sigma Q-Q$. Replacing $Q$ changes this by a coboundary; replacing $P$ by $P+nR$, $R\in E(K)$, does not change it. If its class is zero, some torsion translate of $Q$ is Galois-invariant, so $P\in nE(K)$. This proves injectivity of the <Kummer map of an elliptic curve>.
The whole group $H^1(K,E[n])$ need not be finite. The crucial restriction is ramification. Let $S$ contain the finite <primes> of <bad reduction of an elliptic curve> and those dividing $n$. At $v\notin S$, <good reduction> and prime-to-residue-characteristic multiplication imply that every division point of a local point is in an <unramified extension>, by the formal-group lifting argument. Thus every global Kummer class restricts trivially to inertia there.
Choose a finite Galois extension $L/K$ containing $E[n]$ and the $n$th roots of unity, and enlarge $S$ to include its ramified <primes>. Over $L$, the torsion module is constant and can be identified with $\mu_n^2$. By <Kummer theory>,
$$
H^1(L,E[n])\simeq\bigl(L^{\times}/(L^{\times})^n\bigr)^2.
$$
For classes unramified outside the <primes> $S_L$ above $S$, <valuations> outside $S_L$ must be divisible by $n$. Their restriction images therefore lie in $L(S_L,n)^2$, the square of an <S-unramified power class group>. Its finiteness can be seen without assuming the <Mordell-Weil theorem>: the <finiteness of S-unramified Kummer classes> follows from
$$
0\to\mathcal O_{L,S_L}^{\times}/(\mathcal O_{L,S_L}^{\times})^n
\to L(S_L,n)\to\operatorname{Cl}(\mathcal O_{L,S_L})[n]\to0.
$$
The <S-unit group> is finitely generated by the <Dirichlet unit theorem> and its <valuations> at $S_L$. The localized <ideal class group> is a quotient of the ordinary ideal class group of $L$, which is finite by the <finiteness of the ideal class group>. The <Minkowski bound for ideal classes> supplies an integral ideal of bounded norm in each class, and only finitely many integral ideals have bounded norm.
The kernel of restriction from $H^1(K,E[n])$ to $H^1(L,E[n])$ is also finite: a cocycle trivial after restriction can, after subtracting a coboundary, be made to factor through the finite group $\operatorname{Gal}(L/K)$. There are only finitely many maps from that group to the finite module $E[n]$. This is the <finite-extension kernel of a Kummer map> mechanism. Consequently the possible Kummer classes form a finite set, proving
$$
\boxed{E(K)/nE(K)\text{ is finite for every }n\geq2.}
$$
This is the <Weak Mordell-Weil theorem>. Imposing membership in the local Kummer image at every place refines the finite unramified collection to the <n-Selmer group>. Its relation with the obstruction to a globally <rational point> is
$$
0\to E(K)/nE(K)\to\operatorname{Sel}^{(n)}(E/K)\to\operatorname{Sha}(E/K)[n]\to0,
$$
where the last group is the $n$-torsion of the <Tate–Shafarevich group>. Thus local solubility gives a finite computable upper bound but can leave a genuine global obstruction.
\b[Heights and finite generation.] For the other essay, normalize the <absolute values on a field> $K$ so that the <product formula> holds. The <Absolute logarithmic Weil height> of $[x_0:x_1]\in\mathbb P^1(K)$ is
$$
h([x_0:x_1])=\frac1{[K:\mathbb Q]}\sum_v [K_v:\mathbb Q_v]\log\max\{|x_0|_v,|x_1|_v\}.
$$
At the infinite places the local degrees are one or two and the usual real or complex modulus is used. The <product formula> makes this independent of scaling; it is also unchanged by extending $K$. Over $\mathbb Q$, using <coprime> integral coordinates gives $h([u:v])=\log\max(|u|,|v|)$, the <naive height on the projective line>. The <Northcott theorem> says that points of bounded height and bounded field degree form a finite set; in particular this holds over the fixed field $K$. One way to see the finiteness is the <height-Mahler measure formula>: for an algebraic number of degree $d$, the coefficients of its primitive minimal integer polynomial are bounded by binomial factors times $\exp(dh)$. Bounds on $d$ and $h$ therefore leave only finitely many integer polynomials and hence only finitely many algebraic numbers.
Set $h_E(P)=h(x(P))$ for $P\ne O$, and $h_E(O)=0$. Since $x:E\to\mathbb P^1$ has degree two, bounded $h_E$ gives only finitely many points of $E(K)$. The <height growth under a morphism of the projective line> states that a degree-$d$ morphism gives $h(f(t))=d\,h(t)+O(1)$. The upper bound follows from its homogeneous coordinate polynomials; the lower bound follows from their having no common zero, using a resultant identity at each place. Applied to the degree-four duplication map, it gives a uniform constant $C$ with
$$
|h_E(2P)-4h_E(P)|\leq C.
$$
Consequently $4^{-j}h_E(2^jP)$ is Cauchy: the absolute difference of consecutive terms is at most $C4^{-(j+1)}$. Define the <canonical height of an elliptic curve> by
$$
\widehat h(P)=\frac12\lim_{j\to\infty}4^{-j}h_E(2^jP),\qquad
\left|\widehat h(P)-\frac12h_E(P)\right|\leq\frac C6.
$$
It is nonnegative and satisfies $\widehat h(2P)=4\widehat h(P)$. The factor one-half is the conventional normalization for the divisor $(O)$, since the first-coordinate height belongs to $2(O)$.
The addition-divisor identity also gives the bounded-error height identity
$$
h_E(P+Q)+h_E(P-Q)=2h_E(P)+2h_E(Q)+O(1),
$$
uniformly in $P,Q$. This is the height form of the identity for the sum and difference pullbacks of the <line bundle> associated to $2(O)$; it includes points where the affine addition formulas have a zero denominator. Apply it to $2^jP,2^jQ$, divide by $2\cdot4^j$, and take the limit. Thus $\widehat h$ satisfies the <parallelogram law>, and its polarization is a <bilinear form>, the height pairing. Nonnegativity implies the <Cauchy-Schwarz inequality> for that pairing, by evaluating its nonnegative quadratic polynomial on $mP+nQ$ for all integers $m,n$ and approximating real ratios by rational numbers. In particular
$$
\widehat h(P-Q)\leq2\widehat h(P)+2\widehat h(Q).
$$
These properties do not presuppose finite generation. Bounded canonical height gives a finite set by the bounded difference and the <Northcott theorem>. Torsion points have canonical height zero. Conversely, if $\widehat h(P)=0$, all $2^jP$ have bounded naive height, so two coincide; their difference makes $P$ torsion. Thus the height detects precisely the free part, once finite generation is proved.
Use the <Weak Mordell-Weil theorem> with $n=2$ and choose finitely many coset representatives $R_i$. Write $P=2Q+R_i$, and put $H=\max_i\widehat h(R_i)$. The height inequality yields
$$
4\widehat h(Q)=\widehat h(P-R_i)\leq2\widehat h(P)+2H,
\qquad \widehat h(Q)\leq\tfrac12\widehat h(P)+\tfrac12H.
$$
Iteration enters the finite set $\{P:\widehat h(P)\leq H+1\}$, since after $j$ steps the height is at most $H+2^{-j}\max(\widehat h(P)-H,0)$. Unwinding $P=2Q+R_i$ expresses every point using that finite set and the $R_i$. This proves the <canonical-height proof of Mordell-Weil finite generation> and hence
$$
\boxed{E(K)\cong\mathbb Z^r\oplus E(K)_{\rm tors},\qquad r<\infty,\quad\#E(K)_{\rm tors}<\infty.}
$$
The <Fundamental theorem of finitely generated abelian groups> supplies this decomposition. The <Mordell-Weil theorem> combines a finite quotient from arithmetic <Galois cohomology> with a contracting height descent; either ingredient alone would not establish finite generation.
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