Proof. Since \(E(\mu_n)/E\) is abelian, \(E\) contains every root of unity. Adjoining an \(n\)th root of an element of \(E\) then gives a cyclic extension, so \(E\) is closed under radicals. Finite solvable Galois extensions can be built by towers of radical extensions after adjoining roots of unity; hence \(\mathbb Q^s\subset E\).

The same tower argument shows that \(\mathbb Q^s\) contains all roots of unity and is closed under radicals. Any nontrivial finite abelian extension would have a cyclic subextension of prime degree, which Kummer theory excludes. Any nontrivial abelian algebraic extension would have a nontrivial finite abelian subextension. Thus \(\mathbb Q^s\) is solvably closed.

Finally, a finite solvable Galois extension of \(M\) lies in a finite solvable Galois extension of \(\mathbb Q\): first take a solvable normal closure of \(M/\mathbb Q\) inside \(\mathbb Q^s\), and then the compositum of the finitely many conjugate extensions. The group of this compositum over that normal closure embeds in a product of solvable groups. This proves the last assertion. ◻

Proof. An open subgroup is \(P_M\) for a number field \(M\subset\mathbb Q^s\). An element centralizing \(P_M\) is a field automorphism of \(\mathbb Q^s\) implementing the identity automorphism of \(P_M\). The identity field map also implements it, so uniqueness in Theorem 3 gives the assertion. ◻

Proof. Let \(p\ne\ell\) be the residue characteristic of \(v\), and choose a topological generator \(\tau\) of a pro-\(\ell\) Sylow subgroup of \(I_v\). Its image in source tame inertia generates the pro-\(\ell\) factor. The tame conjugation relation shows that \[e=g\tau g^{-1}\tau^{-(Nv)^{b(g)}}\] belongs to source wild inertia, a pro-\(p\) group. In the target tame quotient, the image of \(e\) also lies in the pro-\(\ell\) factor of tame inertia: both \(\varphi(\tau)\) and its conjugate do. Its image is therefore trivial, because it belongs to a pro-\(p\) subgroup and to a pro-\(\ell\) group. The nontrivial image of \(\varphi(\tau)\) in the torsion-free target pro-\(\ell\) tame factor then gives (1) by the target tame conjugation relation. The target residue characteristic is different from \(\ell\) by Theorem 5. ◻

Proof. The multiplicative hypothesis implies that the cyclic subgroup generated by every \(u^a\) with \(a\ne0\) is Zariski dense in \(\mathbb G_m^n\). Indeed, distinct Laurent monomials have distinct values at \(u^a\). If a Laurent polynomial vanished at all nonnegative powers of \(u^a\), a Vandermonde matrix applied to its finitely many monomials would force all its coefficients to vanish.

Suppose that the logarithms span a subspace of dimension \(r<n\). The case \(r=0\) contradicts the multiplicative hypothesis because the kernel of the logarithm on local units consists of roots of unity. Thus \(1\leq r<n\), and in particular \(n\geq2\). We will obtain an algebraic determinant whose small \(\ell\)-adic absolute value is incompatible with its archimedean size.

First make all choices needed for the analytic estimate. Replacing all \(u_i\) by the same positive integer power preserves the hypotheses and the logarithmic rank. Such a power puts their coordinates in a neighborhood where exponential and logarithm are inverse. The \(\mathcal O_K\)-module generated by their logarithms is free of rank \(r\); choose a basis \(b_1,\ldots,b_r\in K^n\). After a further common \(\ell\)-power, these basis vectors have sufficiently small coordinates that \[\rho:=\max_{j,t}|b_{t,j}|_\ell \quad\hbox{satisfies}\quad \eta:=\rho\ell^{1/(\ell-1)}<1.\] Here the basis is multiplied by the same power as the logarithms. Consequently there are \(z_i\in\mathcal O_K^r\) such that \[\log u_i=\sum_{t=1}^r z_{i,t}b_t.\] From now on these powered points, the number field \(E\), the local field \(K\), the basis, and \(\eta\) are fixed. In particular, none depends on the degree parameter introduced below.

For an integer vector \(\alpha=(\alpha_1,\ldots,\alpha_n)\), the monomial \(X^\alpha\) on these points is represented by \[f_\alpha(z)= \exp\!\left(\sum_{j=1}^n\alpha_j \sum_{t=1}^r b_{t,j}z_t\right).\] It is analytic on a polydisc of radius greater than \(1\). If \(f_\alpha(z)=\sum_{\nu\in\mathbb Z_{\geq0}^r}c_{\alpha,\nu}z^\nu\), the exponential series and \(|k!|_\ell^{-1}\leq\ell^{k/(\ell-1)}\) give the uniform bound \[ |c_{\alpha,\nu}|_\ell\leq\eta^{|\nu|}, \qquad |\nu|=\nu_1+\cdots+\nu_r. \tag{2}\] The bound is independent of \(\alpha\), since every integer has \(\ell\)-adic absolute value at most \(1\). Moreover, for every \(a\in\mathbb Z_{\geq0}^m\), the point \(z(a):=\sum_i a_i z_i\) lies in \(\mathcal O_K^r\), and \(f_\alpha(z(a))=(u^a)^\alpha\).

We next show that sufficiently many of these evaluations have full algebraic rank. For an integer \(D\geq1\), put \(A=(D+1)^n\) and choose an integer \(S\geq1\) such that \[ (S+1)^m>(nD)^n, \qquad S=O(D^{n/m}). \tag{3}\] Consider the matrix with rows indexed by \(\alpha\in\{0,\ldots,D\}^n\), columns indexed by \(a\in\{0,\ldots,nS\}^m\), and entries \((u^a)^\alpha\). We claim that its rank is \(A\).

Otherwise a nonzero polynomial \(P\) of degree at most \(D\) in each variable would vanish at every point \(u^a\) in this grid. Over an algebraic closure of \(E\), define closed subsets of the torus by \[Z_j=\{x\in\mathbb G_m^n: P(u^a x)=0\ \hbox{for all }a\in\{0,\ldots,jS\}^m\}, \qquad 0\leq j\leq n.\] These sets are nested and nonempty, since \(1\in Z_n\), whereas \(\dim Z_0\leq n-1\). Hence some adjacent pair \(Z_j,Z_{j+1}\) has the same dimension \(d\). Let \(V\) be a \(d\)-dimensional irreducible component of \(Z_{j+1}\). For every \(b\in\{0,\ldots,S\}^m\), the translate \(u^bV\) is contained in \(Z_j\) and has dimension \(d\), so it is an irreducible component of \(Z_j\). All these translates are distinct: an equality \(u^bV=u^{b'}V\) for \(b\ne b'\) would make \(V\) stable under every power of \(u^{b-b'}\). The density proved above would then force \(V=\mathbb G_m^n\), contrary to \(V\subseteq Z_0\).

For completeness, a common zero set of polynomials of total degree at most \(\delta\geq1\) in \(\mathbb A^n\) has at most \(\delta^n\) irreducible components. One can obtain this form of the affine Bézout bound by assigning a component \(Y\) the weight \(\delta^{\dim Y}\deg Y\). Start with \(\mathbb A^n\), of weight \(\delta^n\), and impose the equations successively. A component contained in the next hypersurface retains its weight; a component not contained in it is cut into components of dimension one less, whose total degree is at most \(\delta\deg Y\) by Bézout. Thus the total weight does not increase, and removing redundant components can only decrease it. The same component bound holds after restriction to the torus. Applying it to \(Z_j\), whose equations have total degree at most \(nD\), contradicts (3). This proves the rank claim.

Choose a nonzero \(A\times A\) minor \(\Delta_D\) of the evaluation matrix. It is an algebraic integer in the fixed field \(E\). Expand its rows using the convergent power series for \(f_\alpha\). A nonzero term must involve \(A\) distinct monomials in the \(r\) variables \(z_1,\ldots,z_r\), since repeated monomials give repeated rows in the corresponding evaluation determinant. The sum of the degrees of any \(A\) distinct such monomials is at least \(c_rA^{1+1/r}\) for all sufficiently large \(A\): the number of monomials of degree at most \(t\) is \(\binom{t+r}{r}=O_r(t^r)\), so at least half of the \(A\) monomials have degree bounded below by a positive multiple of \(A^{1/r}\). Their evaluations have absolute value at most \(1\). The nonarchimedean triangle inequality and (2), first for truncated series and then by convergence, therefore give a constant \(c>0\) such that \[ \log|\Delta_D|_\ell\leq-cA^{1+1/r}. \tag{4}\]

At every archimedean embedding \(\tau:E\hookrightarrow\mathbb C\), an entry of this minor has absolute value at most \(\exp(CDS)\), with \(C\) independent of \(D\). Expanding the determinant thus gives \[ \log|\tau(\Delta_D)|\leq C'ADS+A\log A. \tag{5}\] Let \(\lambda\) be the place of \(E\) induced by \(E\hookrightarrow K\). Use absolute values at finite places extending the usual ones on \(\mathbb Q\), and sum over all embeddings \(E\hookrightarrow\mathbb C\), so that each complex place occurs twice. Writing \(p_v\) for the rational prime below a finite place \(v\), the product formula reads \[0=\sum_{\tau:E\hookrightarrow\mathbb C}\log|\tau(\Delta_D)| +\sum_{v\text{ finite}}[E_v:\mathbb Q_{p_v}] \log|\Delta_D|_v.\] Every term in the second sum is nonpositive because \(\Delta_D\) is an algebraic integer. Retaining its \(\lambda\)-term and using (4)–(5) would imply \[c[E_\lambda:\mathbb Q_\ell]A^{1+1/r} \leq [E:\mathbb Q](C'ADS+A\log A).\] This is impossible as \(D\to\infty\): the left side has order \(D^{n+n/r}\), whereas the right side is \(O(D^{n+1+n/m}+D^n\log D)\), and \[\frac nr\geq\frac n{n-1}>1+\frac nm\] by \(m>n(n-1)\). This contradiction proves the lemma. ◻

Proof. We first recall the class-field-theoretic description of the character space. Let \(V_M=M\otimes_\mathbb Q\mathbb Q_\ell\), and let \(U_M\subseteq V_M\) be the \(\mathbb Q_\ell\)-linear span of the logarithms of the global units, with logarithms taken at all places above \(\ell\). Then \[ X_{\ell}(M)\simeq \{\lambda\in\mathop{\mathrm{Hom}}_{\mathbb Q_\ell}(V_M,\mathbb Q_\ell): \lambda(U_M)=0\}. \tag{6}\] Indeed, pass to the inverse limit of the ray class sequences with moduli supported above \(\ell\). Local logarithms identify the rationalized local unit groups with \(V_M\); the global units impose exactly the subspace \(U_M\). The ordinary class group and the factors at real places are finite, so contribute no additive \(\mathbb Q_\ell\)-characters. This description uses the actual logarithmic image of the units and makes no assertion about its dimension.

For \(v\nmid\ell\), choose \(h_v>0\) and \(b_v\in\mathcal O_M\) with \((b_v)=v^{h_v}\). Such a choice is possible because the ideal class group is finite. Principal reciprocity, with arithmetic Frobenius, gives for the linear form corresponding to \(x\) the identity \[ \lambda(\log b_v)=-h_vx(\mathop{\mathrm{Fr}}_v). \tag{7}\] Here \(b_v\) is a unit at every place above \(\ell\); all other local unit contributions are killed by \(x\).

For (i), set \(n=[M:\mathbb Q]\) and choose \(m>n(n-1)\) of the zero primes, over distinct rational primes unramified in a Galois closure \(E\) of \(M\). Let \(b_i\) be the elements just chosen for these primes, and let \[u_i=(\sigma(b_i))_{\sigma:M\hookrightarrow E}\in(E^\times)^n.\] All coordinates are algebraic integers and are units at every place of \(E\) above \(\ell\). We verify the multiplicative hypothesis of Lemma 8 by valuations.

Put \(G=\mathop{\mathrm{Gal}}(E/\mathbb Q)\) and \(H=\mathop{\mathrm{Gal}}(E/M)\). For a selected degree-one prime \(v_i\) and a prime \(\mathfrak p_i\) of \(E\) above it, the decomposition group \(D_i\) is contained in \(H\). In fact, the rational prime is unramified in \(E\), and the degree-one condition says that \([D_i:D_i\cap H]=1\). The support of \(v_i\mathcal O_E\) is indexed by \(H/D_i\). Its translates corresponding to the embeddings \(M\hookrightarrow E\), indexed by \(G/H\), have supports indexed by the disjoint sets \(gH/D_i\). Distinct indices \(i\) involve distinct rational primes. It follows that the \(mn\) numbers \(\sigma(b_i)\) have pairwise disjoint nonempty supports of their principal ideals. They are therefore multiplicatively independent modulo roots of unity. In particular, for any nonzero \(a\in\mathbb Z^m\), the \(n\) coordinates of \(u^a\) are multiplicatively independent modulo roots of unity.

Embed \(E\) in a finite extension \(K/\mathbb Q_\ell\). Under the splitting \[(M\otimes_\mathbb Q\mathbb Q_\ell)\otimes_{\mathbb Q_\ell}K \simeq\prod_{\sigma:M\hookrightarrow E}K,\] the vector \(\log b_i\) becomes \(\log u_i\). Lemma 8 says that these vectors span the right side. Equations (6) and (7) therefore force \(\lambda=0\), and hence \(x=0\).

For (ii), put \(C=\langle\delta\rangle\), \(F=M^C\), and \(y=E_Cx\). Since \(\delta\) fixes every indicated prime \(v\), averaging preserves its Frobenius value, so \(y(\mathop{\mathrm{Fr}}_v)=0\). The restriction–corestriction identity gives \[\mathop{\mathrm{res}}_{M/F}\mathop{\mathrm{cor}}_{M/F}y=\sum_{g\in C}g y=|C|y.\] Thus \(y\) is the restriction of \(z=|C|^{-1}\mathop{\mathrm{cor}}_{M/F}y\in X_{\ell}(F)\). At an indicated prime, the decomposition group of \(M/\mathbb Q\) is exactly \(C\). The prime \(u=v\cap F\) consequently has absolute residue degree one, and \[0=y(\mathop{\mathrm{Fr}}_v)=[\kappa(v):\kappa(u)]z(\mathop{\mathrm{Fr}}_u).\] Infinitely many distinct \(u\) occur because \(M/F\) is finite. Part (i) gives \(z=0\), and therefore \(y=0\).

For (iii), suppose that infinitely many zero primes exist. Let \(E/\mathbb Q\) be a finite Galois closure of \(M\), put \(R=\mathop{\mathrm{Gal}}(E/\mathbb Q)\), and let \(y=\mathop{\mathrm{res}}_{E/M}x\). Discarding finitely many primes, choose primes of \(E\) above these zero primes whose rational primes are unramified in \(E\). Restriction multiplies a Frobenius value by the relative residue degree, so they are zero primes for \(y\). An infinite subset has one fixed Frobenius element \(\delta\in R\). By (ii), \(E_{\langle\delta\rangle}y=0\), and hence \(E_Ry=0\). It follows that \[\mathop{\mathrm{res}}_{E/\mathbb Q}\mathop{\mathrm{cor}}_{E/\mathbb Q}y=|R|E_Ry=0.\] Restriction on additive characters across a finite extension is injective: a character that vanishes on a subgroup of finite index has finite image in the torsion-free group \(\mathbb Q_\ell\), and so is zero. On the other hand, transitivity and the corestriction–restriction identity give \[\mathop{\mathrm{cor}}_{E/\mathbb Q}y =\mathop{\mathrm{cor}}_{M/\mathbb Q}\mathop{\mathrm{cor}}_{E/M}\mathop{\mathrm{res}}_{E/M}x =[E:M]\mathop{\mathrm{cor}}_{M/\mathbb Q}x\ne0,\] a contradiction. ◻

Proof. By the normal basis theorem, \(V_M=M\otimes_\mathbb Q\mathbb Q_\ell\) is the regular representation of \(R\), as is its dual. The naturality of (6) gives the second embedding.

The equivariant Dirichlet unit theorem [1] identifies the representation \(\mathcal O_M^\times\otimes_\mathbb Z\mathbb Q_\ell\) with the permutation representation on the archimedean places minus its trivial summand. Since these places are indexed by \(R/\langle c\rangle\), writing \(U^{\mathrm{alg}}=\mathcal O_M^\times\otimes_\mathbb Z\mathbb Q_\ell\) gives \[\mathbb Q_\ell\oplus U^{\mathrm{alg}} \simeq\mathop{\mathrm{Ind}}_{\langle c\rangle}^{R}\mathbf1.\] The character identity follows first over the real numbers from the usual logarithmic unit embedding, and hence over \(\mathbb Q_\ell\) by semisimplicity in characteristic zero. In particular, \(U^{\mathrm{alg}}\) is self-dual. Its \(\ell\)-adic logarithmic image \(U_M\) is a quotient of \(U^{\mathrm{alg}}\); no injectivity of the logarithm on this tensor product is needed.

The decomposition of the regular representation induced from the two characters of \(\langle c\rangle\) is \[\mathbb Q_\ell[R] \simeq\mathop{\mathrm{Ind}}_{\langle c\rangle}^{R}\mathbf1\oplus W_M \simeq\mathbb Q_\ell\oplus U^{\mathrm{alg}}\oplus W_M.\] Since \(U_M^\vee\) is a subrepresentation of \((U^{\mathrm{alg}})^\vee\simeq U^{\mathrm{alg}}\), semisimplicity and \(X_{\ell}(M)\simeq(V_M/U_M)^\vee\) show that \(X_{\ell}(M)\) contains \(\mathbb Q_\ell\oplus W_M\). This proves the first embedding in (8).

Finally, realize \(W_M\) as the direct sum of the signed lines indexed by the left cosets of \(\langle c\rangle\). An element acting trivially must fix the line of the identity coset, and therefore belongs to \(\langle c\rangle\). The nonidentity element \(c\) acts by \(-1\) on that line. Thus only the identity acts trivially on \(W_M\), proving faithfulness. ◻

Proof. Choose a quotient \(R_1=\mathop{\mathrm{Gal}}(L_1/\mathbb Q)\) dominating \(R_0\) such that every \(H_i\) has nontrivial image in \(R_1\), the kernel of \(P\to R_1\) is contained in \(A\), and \(L_1\) contains \(\mathbb Q(i)\). These are finitely many compatible conditions on an open normal subgroup of \(P\). Put \(n=|R_1|\).

We will enlarge \(R_1\) while keeping element orders bounded by \(2n\). At the same time, the images of the \(H_i\) and the conjugacy class of complex conjugation will grow. Growth of the \(H_i\) makes the sum of their invariant spaces small. Growth of the conjugacy class makes the signed representation supply enough \(D\)-invariant vectors killed by \(E_C\) to escape that sum.

We use the regular-module auxiliary extension construction appearing in Uchida’s proof [14], with quantitative bounds on its invariant spaces. More precisely, we construct a further quotient with kernel consisting of \(m\) copies of the regular \(\mathbb F_2[R_1]\)-module, where \(m\) will be chosen below. Choose \(m\) distinct rational primes that split completely in \(L_1\). For each of them, select one prime of \(L_1\) above it. Approximation gives elements \(a_1,\ldots,a_m\in L_1^\times\) such that \(a_j\) has valuation one at its selected prime and valuation zero at every other prime in these \(m\) sets. The \(mn\) elements \[g(a_j),\qquad g\in R_1,\quad 1\leq j\leq m,\] have independent square classes: their valuations at the selected sets of primes form a permutation of the identity matrix modulo two. Adjoin their square roots and call the resulting field \(L\). Kummer theory gives \[ 1\longrightarrow V\longrightarrow R=\mathop{\mathrm{Gal}}(L/\mathbb Q) \longrightarrow R_1\longrightarrow1, \qquad V\simeq\mathbb F_2[R_1]^m \tag{10}\] as \(R_1\)-modules. The dual module supplied by Kummer theory is again the regular permutation module. The field \(L\) is Galois over \(\mathbb Q\) because all conjugates of the \(a_j\) were included; its group is solvable, so \(L\subset\mathbb Q^s\). No splitting of (10) is required.

Every element of \(R\) has order at most \(2n\), since its \(n\)th power lies in the exponent-two group \(V\). Write \(A_R\) and \(H_{i,R}\) for the images of \(A\) and \(H_i\) in \(R\). The choice of \(R_1\) gives \(V\subset A_R\). Choose \(h_i\in H_{i,R}\) with nonidentity image \(g_i\in R_1\), of order \(d_i>1\). As \(H_{i,R}\) is normal in \(A_R\), \[(g_i-1)V=[h_i,V]\subset H_{i,R}.\] On \(m\) regular modules the rank of \(g_i-1\) is \(mn(1-1/d_i)\geq mn/2\). Consequently \[ |H_{i,R}|\geq 2^{mn/2}. \tag{11}\] If \(c\in R\) is a complex conjugation, its image in \(R_1\) is a nontrivial involution because \(L_1\) is totally imaginary. Conjugation by \(V\) alone therefore gives at least \(2^{mn/2}\) distinct conjugates of \(c\); thus \[ |c^R|\geq 2^{mn/2}. \tag{12}\]

Set \(\eta=2^{-mn/2}\) and write \(X=X_{\ell}(L)\). By Proposition 10, \(X\) embeds in the regular \(\mathbb Q_\ell[R]\)-representation and contains \[W=\mathop{\mathrm{Ind}}_{\langle c\rangle}^{R}(\operatorname{sign}).\] The regular-representation bound and (11) give \[ \dim\left(\sum_{i=1}^rX^{H_i}\right) \leq\sum_{i=1}^r\frac{|R|}{|H_{i,R}|} \leq r\eta|R|. \tag{13}\] The character of \(W\) takes the value \(|R|/2\) at the identity, the value \(-|R|/(2|c^R|)\) on \(c^R\), and zero elsewhere. Hence, for every subgroup \(J\subset R\), \[ \left|\dim W^J-\frac{|R|}{2|J|}\right| \leq\frac{\eta|R|}{2}. \tag{14}\] For \(D<C=\langle\gamma\rangle\), the map \(E_C:W^D\to W^C\) is surjective. Since \(|D|\leq |C|/2\) and \(|C|\leq2n\), (14) implies \[\begin{align*} \dim\ker(E_C|_{W^D}) &=\dim W^D-\dim W^C\\ &\geq\frac{|R|}{2}\left(\frac1{|D|}-\frac1{|C|}\right) -\eta|R|\\ &\geq |R|\left(\frac1{4n}-\eta\right). \end{align*}\] Choose the integer \(m\geq1\) so large that \[(r+1)2^{-mn/2}<\frac1{4n}.\] This single choice works for all \(\gamma\) and all proper \(D<C\). The last dimension is then greater than (13), so its kernel contains a vector outside the indicated sum. This vector gives (9). If \(C\) is trivial, there is no proper subgroup to consider. ◻

Proof. Choose \(R\) by Lemma 12 and choose any lift \(\gamma\) of \(\gamma_0\). Put \(C=\langle\gamma\rangle\). Take an open normal subgroup \(U'\trianglelefteq\mathcal G\) contained in \(U\cap\beta^{-1}(P_L)\); for example, take the core of this open subgroup in \(\mathcal G\). Let \(F'/\mathbb Q\) be the finite Galois extension with \(G_{F'}=U'\). The map \[\beta|_{U'}:G_{F'}\longrightarrow P_L\] is open.

Chebotarev’s theorem supplies infinitely many primes \(w'\) of \(L\), over distinct rational primes \(q\ne\ell\) unramified in \(L\), with Frobenius exactly \(\gamma\). Theorem 5 applied to \(\beta|_{U'}\) pairs all but finitely many of them with \(\ell\)-active primes \(v'\) of \(F'\), away from \(\ell\). Uniqueness of the target prime for an active source prime implies that infinitely many \(v'\) occur. Since only finitely many primes of \(F'\) lie over a given rational prime, we may pass to a subset with distinct source rational primes \(p\), all unramified in \(F'\). Passing to another infinite subset, we may also assume that their Frobenius elements in \(\mathop{\mathrm{Gal}}(F'/\mathbb Q)\) are a fixed element \(\delta\).

Fix a source prolongation above each \(v'\), and use it also for the prime \(v=v'|_k\). The original correspondence at \(v'\) can be made with this prolongation: changing the source prolongation conjugates the target group by an element of \(\beta(U')\subset P_L\), which preserves the finite prime \(w'\). Activity passes from \(v'\) to \(v\), since the relevant inertia subgroup for \(U'\) is contained in that for \(T\). Apply Theorem 5 at \(v\) using this same source prolongation. If \(w_1\) is the restriction to \(L\) of its target prolongation, then \[\beta(D_{v'})\subset\beta(D_v)\cap P_L\subset D_{w_1}.\] Uniqueness of the target prime for \(v'\) at the level of \(L\) gives \(w_1=w'\). Consequently the corresponding target rational prime is \(q\), and the image of \(D_v\) in \(R\) lies in \(C\). These changes of prolongations preserve \(v'\) and \(w'\), so they preserve the previously selected finite Frobenius elements \(\delta\) and \(\gamma\).

Put \(H=T/U'\subset\mathcal G/U'\) and let \(f\) be the residue degree of \(v\) over \(p\). Figure 1 records the primes involved. Here \(\delta=\mathop{\mathrm{Fr}}(v'/p)\) and \(\gamma=\mathop{\mathrm{Fr}}(w'/q)\) belong to the finite Galois groups over \(\mathbb Q\). The symbols \(\mathop{\mathrm{Fr}}_{v'}\) and \(\mathop{\mathrm{Fr}}_{w'}\) used below instead denote local arithmetic Frobenius over \(F'\) and \(L\), respectively.

In the decomposition group \(\langle\delta\rangle\subset\mathop{\mathrm{Gal}}(F'/\mathbb Q)\) we have \[ \langle\delta\rangle\cap H=\langle\delta^f\rangle, \qquad f=\min\{j\geq1:\delta^j\in H\}. \tag{15}\] Indeed, the residue degree is the orbit size of the distinguished coset in the action of \(\langle\delta\rangle\) on \(\mathop{\mathrm{Gal}}(F'/\mathbb Q)/H\). This description does not require \(H\) to be normal. In particular \(f\) is constant on the chosen infinite set. The homomorphism \(\beta\) induces \(H\to R\), since \(\beta(U')\subset P_L\); define \[d=\beta(\delta^f)\in R.\] This notation is independent of the lift of \(\delta^f\) to \(T\). A local lift of rational Frobenius to the \(f\)th power is a Frobenius lift at \(v\), so the preceding containment gives \(d\in C\).

We claim that \(D=\langle d\rangle\) equals \(C\). Suppose that \(D<C\), and choose \(x\) as in (9). Pullback along \(\beta|_{U'}\) gives a linear map \[j:X_{\ell}(L)\longrightarrow X_{\ell}(F'),\qquad y=j(x).\] The map \(j\) is injective: its defining homomorphism has open image in \(P_L\), and an additive \(\mathbb Q_\ell\)-valued character vanishing on an open subgroup vanishes everywhere. Frobenius evaluation at \(w'\) is invariant under \(\gamma\), because \(\gamma\) fixes \(w'\) and acts trivially by conjugation on its unramified local quotient. Therefore \[x(\mathop{\mathrm{Fr}}_{w'})=(E_Cx)(\mathop{\mathrm{Fr}}_{w'})=0.\] The character \(x\) is unramified at \(w'\), so this equality says that it vanishes on the entire decomposition group there. It follows that \(y(\mathop{\mathrm{Fr}}_{v'})=0\) for every prime in our infinite set. Proposition 9(ii) now gives \[ E_{\langle\delta\rangle}y=0. \tag{16}\]

To see why this contradicts the choice of \(x\), write \(a=|\langle\delta\rangle|\) and \(Y=j(X_{\ell}(L))\). Pullback is \(T\)-equivariant, where the action on the target character space is through \(\beta\). In particular \(y\) is fixed by \(N\). In the sum defining (16), the powers \(\delta^j\) belonging to \(H\) are exactly those for which \(f\mid j\), by (15). Each such power acts as the identity on \(y\), because \(x\) is \(D\)-invariant. If \(f\nmid j\), choose a lift \(t\) of \(\delta^j\) to \(\mathcal G\). Then \(t\notin T\), and \(\delta^jy\) is fixed by \(tNt^{-1}\). Multiplying (16) by \(a\) thus yields \[0=\frac af\,y+\sum_{\substack{0\leq j<a\\f\nmid j}}\delta^jy, \qquad y\in\sum_{i=1}^rX_{\ell}(F')^{N_i}.\]

It remains to return this membership to the target character space. The finite group \(U/U'\) acts on \(X_{\ell}(F')\), and its subspace \(Y\) is stable. Semisimplicity supplies a \(U\)-equivariant projection \[\pi:X_{\ell}(F')\longrightarrow Y.\] Every \(N_i\) lies in \(U\), so \(\pi\) carries \(N_i\)-invariants into \(Y^{N_i}\). Equivariance and injectivity of \(j\) give \[Y^{N_i}=j\bigl(X_{\ell}(L)^{H_i}\bigr).\] Applying \(\pi\) to the displayed membership and then applying \(j^{-1}\) contradicts the last condition in (9). Only \(U\)-equivariance of the projection is used; no normality of \(T\) in \(\mathcal G\) is needed. Thus \(D=C\), proving the proposition. ◻

Proof. Let \(R_0\) be any finite quotient of \(P\) and let \(\gamma_0\in R_0\). Proposition 13 produces an element \(d\) in the image of \(\beta\) in a further quotient \(R\) such that \(\langle d\rangle=\langle\gamma\rangle\), where \(\gamma\) lifts \(\gamma_0\). Thus \(\gamma\), and hence \(\gamma_0\), belongs to the image. Since \(\beta(T)\) is compact, surjectivity onto every finite quotient implies \(\beta(T)=P\).

Dirichlet’s theorem on primes in arithmetic progressions shows that \(\mathcal L\) is infinite. Fix \(\ell\in\mathcal L\), take \[R_0=\mathop{\mathrm{Gal}}(\mathbb Q(\mu_\ell)/\mathbb Q)\simeq\mathbb F_\ell^\times,\] and choose \(\gamma_0\) to generate this cyclic group. The proposition gives infinitely many active primes \(v\) of \(k\) over distinct rational primes \(p\ne\ell\), for which the image of a Frobenius lift \(\sigma_v\) acts on \(\mu_\ell\) by a generator. Let \(q\) be the corresponding target rational prime and let \(f=[\kappa(v):\mathbb F_p]\), so \(Nv=p^f\). If \(a(\sigma_v)\in\widehat{\mathbb Z}\) is the target residue exponent, Lemma 7 gives \[ p^f=q^{a(\sigma_v)}\quad\text{in }\mathbb Z_\ell^\times. \tag{18}\] Its reduction modulo \(\ell\) says that \(p^f\) generates \(\mathbb F_\ell^\times\). In particular \(\gcd(f,\ell-1)=1\). But \(1\leq f\leq[k:\mathbb Q]\), and every prime divisor of any \(f>1\) in this range divides \(m_0\), hence divides \(\ell-1\). Consequently \(f=1\).

Taking logarithms in (18) gives \[(\log\chi_\ell\circ\beta)(\sigma_v) =\log p =\log\chi_{k,\ell}(\sigma_v).\] The difference is an element of \(X_{\ell}(k)\) vanishing at infinitely many primes of absolute residue degree one. Proposition 9(i) proves (17). ◻

Proof. Fix \(r\in\mathcal L\). Proposition 10 supplies a nonzero character \(\psi\in X_{r}(B)\) on which \(\mathop{\mathrm{Gal}}(B/\mathbb Q)\) acts by its nontrivial character. By Proposition 9(i), \(\psi(\mathop{\mathrm{Fr}}_w)\) is nonzero at all but finitely many degree-one primes \(w\nmid r\). Fix such a \(w\) above a split rational prime \(q\ne r\), and put \[x_0=(\log\chi_r)|_{P_B},\qquad z=\frac{\psi(\mathop{\mathrm{Fr}}_w)}{\log_r q}.\] The denominator is nonzero: the kernel of the logarithm on \(\mathbb Z_r^\times\) consists of roots of unity, and the positive rational integer \(q\) is not a root of unity. Thus \(z\ne0\), and \(\psi-zx_0\) vanishes on the decomposition group at \(w\), since it is unramified there and vanishes on Frobenius.

Consider its pullback \[y=\beta_B^*(\psi-zx_0)\in X_{r}(K).\] The pullback of \(\psi\) transforms by sign under \(G_k/G_K\). Its corestriction to \(k\) is therefore zero: restricting that corestriction back to \(K\) gives the sum of the two translates, and restriction is injective for additive characters with characteristic-zero coefficients. Theorem 14 identifies \(\beta_B^*x_0\) with \(\log\chi_{K,r}\). Consequently \[\mathop{\mathrm{cor}}_{K/\mathbb Q}(y) =-z[K:\mathbb Q]\log\chi_{\mathbb Q,r}\ne0.\] For every \(v\in\mathcal V_w\) away from \(r\), decomposition containment gives \(y(\mathop{\mathrm{Fr}}_v)=0\), regardless of which auxiliary prime makes the pair active. Proposition 9(iii) permits only finitely many such \(v\). There are also only finitely many primes of \(K\) above \(r\), proving the assertion. ◻

Proof. Integral Kummer theory gives \[H^1_{\mathrm{cont}}(G_F,\mathbb Z_\ell(1)) \simeq\varprojlim_n F^\times/(F^\times)^{\ell^n}.\] The inverse-limit assertion follows from the finite-level Kummer isomorphisms; the degree-zero groups are finite and satisfy the Mittag–Leffler condition. Every continuous rational cocycle has bounded image, since \(G_F\) is compact, and hence becomes integral after multiplication by one power of \(\ell\). The same observation for coboundaries gives rationalization of this integral cohomology. At a prime outside \(S\), inertia acts trivially on \(\mathbb Z_\ell(1)\), and its Kummer evaluation is the valuation times the tame \(\ell\)-adic character. These integral inertia homomorphisms are torsion-free. Thus unramifiedness after rationalization is exactly the vanishing of all integral valuation coordinates outside \(S\).

For completeness, put \(U_S=\mathcal O_{F,S}^\times\) and let \[H_n=\{[a]\in F^\times/(F^\times)^{\ell^n}: \operatorname{ord}_v(a)\equiv0\pmod{\ell^n} \text{ for every }v\notin S\}.\] Writing \(\mathop{\mathrm{Cl}}_S(F)\) for the \(S\)-ideal class group, there is an exact sequence \[ 0\longrightarrow U_S/U_S^{\ell^n} \longrightarrow H_n \longrightarrow \mathop{\mathrm{Cl}}_S(F)[\ell^n] \longrightarrow0. \tag{20}\] The last map sends \([a]\) to the ideal class of \(\operatorname{div}_S(a)/\ell^n\), where \(\operatorname{div}_S\) records the valuations outside \(S\). The transition from level \(n+1\) to level \(n\) induces multiplication by \(\ell\) on these obstruction classes. Since \(\mathop{\mathrm{Cl}}_S(F)\) is finite, their inverse limit is zero. It follows that the kernel of all outside-\(S\) valuations in the completed multiplicative group is precisely \(\varprojlim_n U_S/U_S^{\ell^n}\). The argument takes place in the full completed multiplicative group; it does not assume finite valuation support for its elements. Finally \(U_S\) is finitely generated, so rationalizing its completion gives \(U_S\otimes\mathbb Q_\ell\). This proves (19). All the constructions commute with field automorphisms preserving \(S\), giving the asserted equivariance. ◻

Proof. Fix \(\ell\in\mathcal L\) and any finite set \(J\subset\mathcal M_\ell\) of primes away from \(\ell\). For each \(w\in J\), choose \(a_w\in B^\times\) and \(h_w>0\) with \[(a_w)=w^{h_w}.\] Its compatible Kummer cocycles define a class \[\kappa_w\in H^1_{\mathrm{cont}}(P_B,\mathbb Q_\ell(1)).\] They factor through \(P_B\) because \(\mathbb Q^s\) contains the roots of unity and all the required radicals. Valuations at the distinct primes \(w\) show that these classes are linearly independent. Each is unramified at every finite prime other than \(w\) and the primes above \(\ell\).

Write \[\rho=\chi_\ell\circ\beta_B,\qquad \epsilon=\rho\chi_{K,\ell}^{-1}.\] By Theorem 14, \(\epsilon\) takes values in \(\mu_{\ell-1}\subset\mathbb Q_\ell^\times\). Let \(K'/K\) be the cyclic extension fixed by its kernel, and let \(\Delta=\mathop{\mathrm{Gal}}(K'/K)\). Pullback along the surjection \(\beta_B\) is injective on \(H^1\), with pulled-back coefficient module \(\mathbb Q_\ell(\rho)\). Restriction to \(G_{K'}\) remains injective: its kernel is a first cohomology group of the finite group \(\Delta\), which vanishes in characteristic zero. We therefore obtain independent classes \[c_w\in H^1_{\mathrm{cont}}(G_{K'},\mathbb Q_\ell(1)) \qquad(w\in J).\] Under the natural \(\Delta\)-action on this cohomology, they all lie in the \(\epsilon^{-1}\) eigenspace. Indeed the original coefficient module is \(\mathbb Q_\ell(1)\otimes\epsilon\), so invariance for its twisted action becomes the eigenvalue \(\epsilon^{-1}\) after restriction.

We claim that every \(c_w\) is unramified away from \(\ell\). Suppose that it is ramified at a prime \(u\nmid\ell\) of \(K'\), and let \(v\) be the prime below it in \(K\). The cyclotomic module is trivial on \(I_u\), so the restricted cocycle is a nonzero continuous homomorphism from \(I_u\) to the additive group \(\mathbb Q_\ell(1)\). Choose a pro-\(\ell\) Sylow subgroup \(J_u\subset I_u\) on which it is nonzero, and a pro-\(\ell\) Sylow subgroup \(J_v\subset I_v\) containing \(J_u\). Then \(\beta_B(J_v)\ne1\), so \(v\) is active at \(\ell\). Theorem 5 sends \(J_v\) into inertia at a corresponding prime \(t\nmid\ell\) of \(B\). Since \(t\) occurs in an active pair, \(t\notin J\), whereas \(\kappa_w\) is unramified at \(t\). Inertia at \(t\) acts trivially on the coefficient module, so the Kummer cocycle itself vanishes on that inertia group. Its pullback therefore vanishes on \(J_u\), a contradiction. This proves the claim.

Let \(S\) consist of the archimedean places of \(K'\) and its primes above \(\ell\). By Lemma 16, the classes \(c_w\) give independent elements of the \(\epsilon^{-1}\) eigenspace in \(\mathcal O_{K',S}^{\times}\otimes\mathbb Q_\ell\). It remains to bound this eigenspace. The equivariant Dirichlet \(S\)-unit theorem [1] gives the character of the full \(S\)-unit module as \[[\mathcal O_{K',S}^{\times}\otimes\mathbb Q_\ell] =[\mathbb Q_\ell[S]]-[\mathbb Q_\ell].\] Each \(\Delta\)-orbit in \(S\) contributes at most one copy of the one-dimensional character \(\epsilon^{-1}\) to the permutation representation: for a stabilizer \(H\subset\Delta\), its multiplicity in \(\mathbb Q_\ell[\Delta/H]\) is one if it is trivial on \(H\), and zero otherwise. There are \[\#(S/\Delta) =r_1(K)+r_2(K)+\#\{v\text{ of }K:v\mid\ell\} \le2[K:\mathbb Q]\] such orbits. Subtracting the trivial representation cannot increase this multiplicity. Hence the independent classes \(c_w\) lie in a space of dimension at most \(2[K:\mathbb Q]\), and \(\#J\le2[K:\mathbb Q]\). Since \(J\) was arbitrary, this bounds all missed primes away from \(\ell\). There are at most two primes of \(B\) above \(\ell\), giving the second bound. ◻

Proof. Choose distinct auxiliary primes \(s_j\) larger than \(m\), \(d\), and every prime in the lists, and put \[C=\mathbb Q(\zeta_{m\prod_j s_j}).\] For each \(j\), the classes of the distinct rational primes in \(\{q_j\}\cup\mathcal P_j\) are independent in \(C^\times/(C^\times)^{s_j}\). Indeed, at any prime in those lists, the ramification index in \(C/\mathbb Q\) divides \([\mathbb Q(\zeta_m):\mathbb Q]\): the other cyclotomic conductors are prime to that prime. This index is prime to \(s_j\), and valuations at the separate rational primes give the asserted independence.

Kummer theory therefore identifies the Galois group over \(C\) of \[E_j=C\bigl(q_j^{1/s_j},\ p^{1/s_j}\ (p\in\mathcal P_j)\bigr)\] with an elementary abelian \(s_j\)-group having one independent coordinate for each listed prime. Choose \(\sigma_j\in\mathop{\mathrm{Gal}}(E_j/C)\) which fixes \(q_j^{1/s_j}\) and acts nontrivially on every \(p^{1/s_j}\) with \(p\in\mathcal P_j\). The extensions \(E_j/C\) are linearly disjoint, since their degrees are powers of distinct primes. Thus these automorphisms combine to an element \(\sigma\in\mathop{\mathrm{Gal}}(E/C)\), where \(E\) is their compositum.

The extension \(E/\mathbb Q\) is Galois. Moreover, conjugation in \(\mathop{\mathrm{Gal}}(E/\mathbb Q)\) preserves whether each of the specified radical coordinates of \(\sigma\) is trivial: a conjugation only raises its root-of-unity multiplier to a power prime to the corresponding \(s_j\). Chebotarev’s theorem supplies infinitely many rational primes \(\ell\) with Frobenius in the conjugacy class of \(\sigma\), avoiding the prescribed finite set, all listed primes, and \(2\). They split completely in \(C\), so \(\ell\equiv1\pmod m\) and \(\ell\equiv1\pmod{s_j}\). At these primes, \(q_j\) is an \(s_j\)th power modulo \(\ell\) and every \(p\in\mathcal P_j\) is not.

The quotient of \(\mathbb F_\ell^\times\) by its \(s_j\)th powers has prime order \(s_j\). A nontrivial class stays nontrivial on raising it to any exponent \(1\le f\le d<s_j\). Every power of \(q_j\), however, is an \(s_j\)th power. This proves the displayed separation. In particular, overlaps between different lists impose no conflict: their conditions belong to linearly disjoint Kummer extensions with different prime exponents. ◻

Proof. Suppose that there are infinitely many exceptions. Lemma 15 allows us to choose \[w_1,\ldots,w_n,\qquad n>2[K:\mathbb Q]+2,\] among them, above distinct split rational primes \(q_1,\ldots,q_n\), with every \(\mathcal V_{w_j}\) finite. Let \(\mathcal P_j\) be the set of rational residue characteristics of the primes in \(\mathcal V_{w_j}\). The failure of the asserted conclusion means \(q_j\notin\mathcal P_j\).

Apply Lemma 18 with \(m=m_0\) and \(d=[K:\mathbb Q]\). It gives \(\ell\in\mathcal L\), away from all the primes involved, such that \[ Nv\bmod\ell\notin\langle q_j\bmod\ell\rangle \qquad(v\in\mathcal V_{w_j},\ 1\le j\le n), \tag{21}\] because \(Nv=p^f\) with \(p\in\mathcal P_j\) and \(1\le f\le[K:\mathbb Q]\). If \(w_j\) occurred in an active pair at this \(\ell\), its source prime would lie in \(\mathcal V_{w_j}\). Applying Lemma 7 to a lift of source Frobenius and reducing modulo \(\ell\) would give \[Nv\bmod\ell\in\langle Nw_j\bmod\ell\rangle =\langle q_j\bmod\ell\rangle,\] contrary to (21). All \(n\) primes \(w_j\) therefore belong to \(\mathcal M_\ell\), contradicting Proposition 17. ◻