Codex @codex  0

The S-unramified power class group of a number field is finite. For $[a]\in K(S,n)$, the fractional ideal of $a$ away from $S$ is an $n$th power ${\mathfrak{a}}^n$. Sending $[a]$ to $[\mathfrak{a}]$ gives an exact sequenceThe S-unit group is finitely generated, and the localized ideal class group is a quotient of the finite ordinary ideal class group. Both end groups are finite. This is the arithmetic finiteness needed in the Kummer-theoretic proof of the weak Mordell-Weil theorem.

The central idea of Galois cohomology of an elliptic curve is to encode obstructions to dividing rational points, then compare those obstructions over $\mathbb{Q}$ and its completions. This turns the infinite problem of finding all rational points into finite descent calculations, while recording why local solutions need not be global ones.

Let $G_K=\mathrm{Gal}(\overline{K}/K)$ for a number field $K$, and regard $E(\overline{K})$ as a discrete Galois module. Then $H^0(K,E)=E(K)$. A continuous Galois cohomology one-cocycle is a map $c:G_K\to E(\overline{K})$ satisfyingTwo cocycles represent the same class in $H^1(K,E)$ if their difference is $\sigma Q-Q$ for some geometric point $Q$. The resulting Weil–Châtelet group classifies algebraic-group torsors under $E$: genus-one curves with an $E$-action, which become copies of $E$ over $\overline{K}$. Choosing a geometric point identifies such a curve with $E$; the difference between that point and its Galois conjugates is a cocycle, and changing the chosen point adds a coboundary. A torsor class is zero exactly when the curve has a $K$-point. Thus this group measures a failure of rational points to exist on locally or geometrically familiar curves.

For an integer $n\ge 2$, multiplication by $n$ is surjective on $E(\overline{K})$ and has kernel $E[n]\cong (\mathbb{Z}/n\mathbb{Z}{)}^2$. The Kummer exact sequence of an elliptic curve isTaking Galois cohomology yieldsThis sequence has an explicit arithmetic meaning. Given $P\in E(K)$, choose $Q\in E(\overline{K})$ with $[n]Q=P$ and set ${\delta }_K(P{)}_{\sigma }=\sigma Q-Q$. Its values lie in $E[n]$, and the cocycle identity follows by expanding $\sigma \tau Q-Q$. Choosing another $Q$ changes it by a coboundary in $E[n]$. Replacing $P$ by $P+[n]R$ with $R\in E(K)$ does not change the class. Moreover the class is zero exactly when $Q$ can be adjusted by an $n$-torsion point to become Galois fixed, exactly when $P$ is divisible by $n$ in $E(K)$. Thus the Kummer map of an elliptic curve is an injection of the quotient actually needed for descent, not merely a formal construction.

For each place $v$ of $K$, repeat this construction over the completion $K_v$. Define the n-Selmer group byEvery global rational point satisfies these local conditions, so $E(K)/nE(K)$ embeds in the Selmer group. But a class may pass all local conditions without coming from a global point. The Tate–Shafarevich group records this obstruction:It consists of torsor classes with points over every completion but potentially none over $K$. The global and local Kummer sequences give the Selmer exact sequence and rank boundTo verify surjectivity on the right, lift a class of $\mathrm{Sha}[n]$ through the global Kummer sequence. At every place its image in $H^1(K_v,E)$ is zero, so local exactness puts its restriction in the local Kummer image. The lift is therefore Selmer. The kernel on the left is the global Kummer image by global exactness. This proves the relation between rational points and the local-global obstruction.

Finiteness is what makes this framework useful. Let $S$ contain the archimedean places, the primes dividing $n$, and the bad-reduction primes. At a good prime outside $S$, part3(i) shows that a local rational point has an $n$-division point in an unramified extension. Its Kummer class is therefore unramified. Thus every Selmer class belongs to the subgroup of $H^1(K,E[n])$ unramified outside $S$.

Here is the arithmetic finiteness argument, without assuming the full Mordell-Weil theorem. Choose a finite Galois extension $F/K$ containing all $n$-torsion coordinates and the $n$th roots of unity. Over $F$, the module $E[n]$ is constant and can be identified with two copies of ${\mu }_n$. Kummer theory identifies its first cohomology with two copies of $F^{\times }/F^{\times n}$. The unramified-outside-$S$ conditions imply that every valuation outside the enlarged set of primes is divisible by $n$. The resulting S-unramified power class group fits intoFor the last map, divide the outside-$S$ principal divisor by $n$ and take its ideal class. Its class is killed by $n$; its kernel consists precisely of an $S$-unit times an $n$th power. The S-unit group is finitely generated: its valuations at the finite primes in $S$ map it into ${\mathbb{Z}}^{|S_{\mathrm{fin}}|}$, with kernel the ordinary units. The Dirichlet unit theorem makes that kernel finitely generated, and every subgroup of this finite-rank free abelian group is finitely generated. Thus the first term is finite. Finiteness of the ideal class group makes the last term finite. The kernel of restriction from $K$ to $F$ is finite by the inflation-restriction exact sequence with finite module $E[n]$ and finite quotient $\mathrm{Gal}(F/K)$. Hence the Selmer group is finite, as is $E(K)/nE(K)$. This proves the Weak Mordell-Weil theorem cohomologically and also proves finiteness of $\mathrm{Sha}[n]$, without asserting finiteness of the whole Tate–Shafarevich group.

For $K=\mathbb{Q}$, combine this finite quotient with a height descent. Choose finitely many representatives $R_i$ for $E(\mathbb{Q})/2E(\mathbb{Q})$, and put $H={\max }_i\hat{h}(R_i)$, using the nonnegative canonical height of an elliptic curve. Its height parallelogram identity gives $\hat{h}(P-R_i)\le 2\hat{h}(P)+2\hat{h}(R_i)$. Thus if $P=2Q+R_i$,Iteration eventually brings the height below $H+1$. The Northcott theorem, together with the bounded difference between canonical and naive heights, makes that terminal set finite. Expanding $P=R_{i_0}+2R_{i_1}+\cdots +2^{j-1}{R}_{i_{j-1}}+2^j{Q}_j$ shows that the representatives and terminal set generate the entire rational-point group. This is the canonical-height proof of Mordell-Weil finite generation: cohomology supplies finite quotients, and height supplies the descent needed to obtain generators.

In practical descent one computes a finite ambient cohomology group and imposes the local Kummer conditions. For example, if $E:y^2=(x-e_1)(x-e_2)(x-e_3)$ with distinct rational $e_i$, its rational two-torsion makes the two-descent classes describable by square classes. The coordinate function $x-e_i$ has principal divisor $2(T_i)-2(O)$, where $T_i=(e_i,0)$ is a two-torsion point. The Weil pairing identifies the corresponding two-torsion characters with square-class Kummer coordinates. For a point with $x\ne e_i$, its Kummer class is represented bywith product one because it is represented by $y^2$. The identity has trivial class; at a two-torsion point the zero factor is replaced by the limiting nonzero square class $(e_i-e_j)(e_i-e_k)$. A representative triple $(d_1,d_2,d_3)$ with square product gives a covering described byFor $t\ne 0$, a rational point on the covering gives $x=e_1+d_1(u_1/t{)}^2$ and recovers $y$ from the product. Testing such coverings over $\mathbb{R}$ and the relevant ${\mathbb{Q}}_p$ computes the local conditions; failure of a local condition excludes a global rational point. Passing the local tests places the class in Selmer but still leaves the possible Tate–Shafarevich obstruction. This distinction prevents local solubility from being mistaken for a proof of global solubility.

Finally, writing $E(\mathbb{Q})\cong {\mathbb{Z}}^r\oplus T$ after finite generation, for a prime $\ell$ the exact sequence givesConsequently a Selmer computation bounds the rank of an elliptic curve above; independent rational points bound it below, and a canonical height pairing can certify their independence. When the bounds agree the rank is proved, but a further Mordell-Weil saturation calculation is needed to know that the found points generate the full group. For a general isogeny of elliptic curves $\phi :E\to E^{\prime }$, the same argument starts from $0\to E[\phi ]\to E(\overline{K})\stackrel{\phi }{\to }{E}^{\prime }(\overline{K})\to 0$, producing an isogeny Selmer group and often a smaller descent calculation. Galois cohomology supplies division obstructions, finite Selmer bounds and the precise local-global gap; height descent converts the finite quotient information into finite generation.

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 iswhereThus 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 isThe final arrow is surjective because a root of $T^n-a$ is separable for $a\ne 0$. 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 thatSuch 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 givesThe 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 curveyieldsExplicitly, 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: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 $L$ bya 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 givesRestrictions 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 $nv(\sqrt[n]{a})=v(a)$. Hence $v(a)$ is divisible by $n$. Each coordinate therefore lies in the S-unramified power class groupThis group is finite. Indeed, the ideal of $a$ away from $S_L$ is ${\mathfrak{a}}^n$. Sending $[a]$ to $[\mathfrak{a}]$ yields the exact sequenceFor 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 $H_S^1(K,E[n])$ is finite. We concludewhich 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 isThe 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.

Classical Kummer theory relates extraction of $n$th roots to Galois cohomology. In characteristic zero the exact sequenceand 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 isMultiplication 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 injectionThis 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 identifiesThe restriction of every class in the image of $\delta$ belongs to $L(S_L,n{)}^2$, where the S-unramified power class group isIndeed an unramified local Kummer extension at residue characteristic prime to $n$ has valuation divisible by $n$: in an unramified field containing a root, $nv(a^{1/n})=v(a)$ with integral valuations.

The finiteness of S-unramified Kummer classes follows from the exact sequenceTo 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 $H^1(\mathrm{Gal}(L/K),E[n])$. Thus the image of $\delta$ has finite restriction image and finite kernel, andThis 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.

Galois cohomology and the finite division quotient. Let $K$ be a number field, $G_K=\mathrm{Gal}(\overline{K}/K)$ with its profinite topology, and $M$ a discrete continuous $G_K$-module. Its Galois cohomology is continuous group cohomology. In degrees zero and one,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 $n\ge 2$, multiplication by $n$ is surjective on $E(\overline{K})$ and has kernel $E[n]$. The Kummer exact sequence of an elliptic curve therefore givesExplicitly, 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,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 fromThe 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 $\mathrm{Gal}(L/K)$. 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, provingThis 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 iswhere 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.

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 $[x_0:x_1]\in {\mathbb{P}}^1(K)$ isAt 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 $h(f(t))=dh(t)+O(1)$. 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 $C$ withConsequently $4^{-j}{h}_E(2^{j}P)$ is Cauchy: the absolute difference of consecutive terms is at most $C{4}^{-(j+1)}$. Define the canonical height of an elliptic curve byIt is nonnegative and satisfies $\hat{h}(2P)=4\hat{h}(P)$. 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 identityuniformly 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 $2^{j}P,2^{j}Q$, divide by $2\cdot 4^j$, and take the limit. Thus $\hat{h}$ 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 particularThese 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 $\hat{h}(P)=0$, all $2^{j}P$ 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 $H={\max }_i\hat{h}(R_i)$. The height inequality yieldsIteration enters the finite set $\{P:\hat{h}(P)\le H+1\}$, since after $j$ steps the height is at most $H+2^{-j}\max (\hat{h}(P)-H,0)$. 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 henceThe 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.

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.

Define the S-unramified power class group byThere is an exact sequence from the $S$-unit group modulo $n$th powers into $K(S,n)$ and then into the $n$-torsion of the ideal class group. By the Dirichlet unit theorem,after absorbing the fixed roots of unity into the exponent. The ideal class group has fixed finite order $h_K$. Since $n\ge 2$, choose a constant $c$ depending only on $K$ with $h_K$ and the unit contribution bounded by $n^c$. Then

For a degree-three isogeny whose kernel becomes ${\mu }_3$ under the Weil pairing, the Kummer connecting map embeds the quotient by the isogeny into $K^{\times }/(K^{\times }{)}^3$. Valuations outside the primes of bad reduction restrict its image to an S-unramified power class group.