Results on solvably closed Galois groups, logarithm ranks, additive characters and Kummer classes, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The maximal prosolvable extension — Proposition 1.3. Q^s is solvably closed and closed under radicals, lies in every solvably closed algebraic extension of Q, and contains every finite solvable Galois extension of a number field M in Q^s; abelian and additive characters of G_M factor through Gal(Q^s/M).
• Uchida's conjecture (OpenAI) — Theorem 2.1. For number fields F_1, F_2 and Galois extensions E_i/F_i with E_i solvably closed, every continuous homomorphism Gal(E_1/F_1) -> Gal(E_2/F_2) with open image is induced by a unique field embedding E_2 -> E_1. Proved in the eighth lesson; assumes the named inputs (U1), (U2) and (H).
• Trivial centralizers in Gal(Q^s/Q) — Lemma 3.1. The only element of P = Gal(Q^s/Q) commuting with every element of an open subgroup of P is 1. Assumes (U1).
• Sylow subgroups of profinite groups — Proposition 4.1. Pro-l Sylow subgroups exist and are pro-l; every closed pro-l subgroup lies in one; any two are conjugate; surjections map them onto Sylow subgroups; they meet a closed normal pro-p subgroup (p != l) trivially. No hypothesis.
• Local Galois groups inside the prosolvable closure — Proposition 5.3. A decomposition group in G_M is isomorphic to the absolute Galois group of the completion, compatibly with inertia; for M in Q^s, restriction maps it isomorphically onto the decomposition group in Gal(Q^s/M). No hypothesis.
• The norm identity — Lemma 6.1. For an active pair (v, w) at an odd prime l, (Nv)^b(g) = (Nw)^a(g) in Z_l^x for every g in D_v, with b and a the source and target residue exponents. Assumes the pair given by (U2).
• Cutting by a hypersurface — Proposition 4.1. For a homogeneous prime p with dim S/p >= 1 and a homogeneous F not in p of degree d, every minimal prime of p + (F) has dimension one less, and the multiplicities of those other than the irrelevant ideal sum to at most d e(p). No hypothesis.
• The component bound — Theorem 5.2. The zero set in k^n of polynomials of total degree at most d has at most d^n irreducible components, and so does its intersection with the torus. No hypothesis.
• The l-adic logarithm rank theorem — Theorem 2.1. If m > n(n-1) points u_i of (E^x)^n have coordinates that are algebraic integers and units of a finite extension K of Q_l, and every nontrivial product u^a has coordinates multiplicatively independent modulo roots of unity, then the logarithms log u_i span K^n. No hypothesis.
• A zero estimate on the torus — Proposition 4.1. If (S+1)^m > (nD)^n, the matrix of values (u^a)^alpha, alpha in {0,...,D}^n and a in {0,...,nS}^m, has rank (D+1)^n. No hypothesis.
• Transfer of additive characters — Lemma 2.1. Corestriction of additive characters is a well-defined homomorphism; cor(res x) = [G:H] x; for H normal, res(cor y) is the sum of the conjugates of y; corestriction is transitive; restriction is injective. No hypothesis.
• The class-field description of additive characters — Proposition 3.1. X_l(M) is isomorphic to the linear forms on M (x) Q_l that vanish on the l-adic logarithms of the global units, compatibly with automorphisms, and lambda_x(log b) = -h x(Fr_v) when (b) = v^h. No hypothesis.
• Frobenius detection (OpenAI) — Proposition 4.1. An additive l-adic character vanishing at Frobenius at infinitely many primes of absolute residue degree one is zero; with a fixed Frobenius class delta in a Galois M/Q one gets E_<delta> x = 0; a character with nonzero corestriction to Q vanishes at only finitely many Frobenius elements. No hypothesis.
• The signed representation — Proposition 5.3. For M/Q finite, Galois and totally imaginary with group R and complex conjugation c, there are R-equivariant injections Q_l + Ind_<c>^R(sign) -> X_l(M) -> Q_l[R]; R acts faithfully and dim X_l(M)^J <= |R|/|J|. No hypothesis.
• Extension to the normalizer — Lemma 1.1. The restriction of an open homomorphism G_F -> P to an open normal subgroup U of G_Q extends uniquely to a continuous homomorphism beta on the normalizer G_k of its kernel, and beta is open. Assumes (U1).
• Quotient amplification (OpenAI) — Lemma 2.1. For an odd prime l and a finite quotient R_0 of P there is a totally imaginary Galois L/Q in Q^s, with group R mapping onto R_0, such that for every cyclic C in R and proper subgroup D of C some x in X_l(L)^D has E_C x = 0 and lies outside the invariants of the images of the conjugate kernels. No hypothesis.
• Frobenius generation (OpenAI) — Proposition 3.1. For a finite quotient of P and an element gamma_0 there is a larger finite quotient R and a lift gamma such that the Frobenius images at infinitely many active primes of k generate <gamma>. Assumes (U1) and (U2).
• Surjectivity and cyclotomic logarithms (OpenAI) — Theorem 4.1. beta: G_k -> P is surjective, and log chi_l o beta = log chi_{k,l} for every odd prime l = 1 mod [k:Q]!. Assumes (U1) and (U2).
• l-adic Kummer theory — Proposition 2.2. Kummer theory gives isomorphisms from the l-adic completion of F^x to H^1(G_F, Z_l(1)), and after tensoring with Q_l to H^1(G_F, Q_l(1)), compatible with field isomorphisms. No hypothesis.
• Unramified Kummer classes — Proposition 3.2. Kummer theory identifies O_{F,S}^x (x) Q_l with the classes in H^1(G_F, Q_l(1)) unramified at every finite prime outside S, compatibly with automorphisms preserving S. No hypothesis.
• The equivariant S-unit theorem — Proposition 4.1. For a finite Galois extension K'/K with group Delta and a Delta-stable set S of places containing the archimedean ones, (O_{K',S}^x (x) Q_l) + Q_l is the permutation module Q_l[S]; the eigenspace of a character Delta -> Q_l^x has dimension at most the number of Delta-orbits on S. No hypothesis.
• Finite fibres (OpenAI) — Lemma 2.1. For all but finitely many primes w of Q(i) above split rational primes, only finitely many primes of K form an active pair with w at some auxiliary prime. Assumes (U1) and (U2).
• A uniform bound on missed primes (OpenAI) — Proposition 3.1. For every auxiliary prime l in the infinite set of the fifth lesson, at most 2[K:Q] + 2 primes of Q(i) occur in no active pair at l. Assumes (U1) and (U2).
• Separation by power residues — Lemma 4.1. Given distinct primes q_j and finite sets P_j of primes not containing q_j, there are infinitely many odd primes l = 1 mod m, outside any finite set, with p^f mod l outside <q_j mod l> for all p in P_j and 1 <= f <= d. No hypothesis.
• Matching residue characteristics (OpenAI) — Proposition 5.1. For all but finitely many primes w of Q(i) above split rational primes q, some prime of K above the same q forms an active pair with w at some auxiliary prime. Assumes (U1) and (U2).
• Identification of the normalizer extension (OpenAI) — Proposition 1.1. beta equals restriction G_k -> P followed by conjugation by an element of P. Assumes (U1) and (U2).
• Prosolvable rigidity (OpenAI) — Theorem 2.1. Every continuous homomorphism G_F -> Gal(Q^s/Q) with open image is conjugate in Gal(Q^s/Q) to restriction. Assumes (U1) and (U2).

Reading order

Eight lessons, to be read in order; the second and third (the component bound and the logarithm rank theorem) and the sixth (Kummer classes and S-units) need only the first lesson and can be read independently. The proof of the conjecture assumes Uchida's isomorphism theorem (U1), Uchida's partial local correspondence (U2) and Hoshi's criterion (H), stated in the first lesson; everything else is proved in the programme.