Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 18H (Galois Theory)

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.

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.