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Let $L=F({\mu }_n)$. Its Galois group embeds into $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, by sending an automorphism to the exponent in its action on ${\zeta }_n$. It is therefore finite abelian, of order dividing $\phi (n)$. The finite case of the Fundamental theorem of finitely generated abelian groups supplies a subgroup chain whose successive quotients are cyclic of prime order $\ell$. Every such $\ell <n$, since $\ell \le \phi (n)<n$ for $n>1$ (and for $n=2$ the possible prime is $2$, whose roots of unity already lie in every characteristic-zero field). Thus every intermediate base contains ${\mu }_{\ell }$, because $F$ contains all roots of unity of orders smaller than $n$.

Each cyclic degree-$\ell$ step $E^{\prime }/E$ is a Kummer extension. Here is the relevant criterion explicitly. For a generator $\sigma$ and a primitive $\ell$th root $\zeta \in E$, the operator ${\sum }_{j=0}^{\ell -1}{\zeta }^{-j}{\sigma }^j$ is not identically zero, by linear independence of distinct field automorphisms. Choose its nonzero value $a$; then $\sigma (a)=\zeta a$, so $a^{\ell }\in E$ and $a\notin E$. Since the degree is prime, $E^{\prime }=E(a)$ is obtained by adjoining an $\ell$th radical. Applying this to the subgroup chain proves $F({\mu }_n)/F$ is a succession of Kummer extensions.

Finally use strong induction on $n$, for arbitrary characteristic-zero base fields. Successively adjoining the roots ${\zeta }_1,\dots ,{\zeta }_{n-1}$ is contained in a Kummer tower by the induction hypothesis, applied over each current base. The just-proved assertion then adjoins ${\zeta }_n$ by further Kummer steps. Hence any cyclotomic extension is contained in a Kummer tower. In the given cyclotomic-and-Kummer tower for a soluble extension, replace every cyclotomic step by such a tower; enlarge subsequent steps by compositum with the new intermediate fields. Base change preserves radical adjunction and the requisite roots of unity, possibly making a step trivial. Thus every finite soluble extension is contained in a succession of Kummer extensions.

Choose the radicals with positive imaginary part. The roots of unity areThe first comes from $z^2+z+1=0$. For the second, $t={\zeta }_5+{\zeta }_5^{-1}$ obeys $t^2+t-1=0$, so $t=(\sqrt{5}-1)/2$; solving $z^2-tz+1=0$ gives the displayed expression. The Kummer extension towers areEach step is quadratic and the base contains the second roots of unity. The second tower really has degrees $2,2$: its intermediate field is real, while the added radical is nonreal, and $\sqrt{5}=1+2({\zeta }_5+{\zeta }_5^{-1})$ belongs to the cyclotomic field.

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.

Let $\sigma (\alpha )={\xi }_d\alpha$ with $\alpha \ne 0$. Thenand these $d$ elements are distinct because ${\xi }_d$ is a primitive root of unity. Thus the stabilizer of $\alpha$ in the cyclic Galois group is trivial. The orbit-stabilizer form of the Fundamental theorem of Galois theory givessoMoreover,The fixed field of the whole Galois group is $K$, and thereforeThis is the basic eigenvector mechanism behind a cyclic Kummer extension.