Open homomorphisms of solvably closed Galois groups

Uchida's conjecture: 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. An l-adic logarithm rank theorem shows that Frobenius values at infinitely many primes of degree one determine an additive character; regular Kummer quotients and Frobenius generation make the homomorphism surjective and compatible with cyclotomic logarithms; a uniform bound on missed primes and power-residue separation match residue characteristics, and the homomorphism is identified with conjugated restriction. The course is an exposition of OpenAI's proof (2026); it uses two theorems of Uchida and a criterion of Hoshi without proof.

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Solvably closed fields and Uchida's conjecture
2. Counting irreducible components
3. A logarithm rank lemma
4. Additive characters and Frobenius detection
5. Normalizer extension and Frobenius generation
6. Kummer classes and S-units
7. Matching residue characteristics
8. Identifying the homomorphism

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Associated primes and primary decomposition (Commutative algebra for geometry)
• Dimension theory of Noetherian local rings (Commutative algebra for geometry)
• Graded modules and Hilbert–Samuel functions (Commutative algebra for geometry)
• Krull dimension and Noether normalization (Commutative algebra for geometry)
• The Nullstellensatz and Jacobson rings (Commutative algebra for geometry)
• Decomposition of primes in extensions · Number fields
• Dirichlet's unit theorem · Number fields
• Cyclotomic polynomials and their automorphisms (Class field theory)
• Global existence and the idèlic class field correspondence · Class field theory
• Hilbert's Theorem 90 and Kummer theory · Class field theory
• Profinite groups and infinite Galois theory · Class field theory
• Ray class fields, conductors and ideal reciprocity · Class field theory
• The Chebotarev density theorem · Class field theory
• The global reciprocity law · Class field theory
• Dirichlet's theorem on primes in arithmetic progressions · Dirichlet L-functions and primes in progressions
• Hensel's lemma, squares and roots of unity in p-adic fields · Local fields
• Places of number fields in extensions and the product formula · Local fields
• Tame ramification and the tame Galois group · Local fields
• Ramification groups and the different of a local extension · Local fields
• The multiplicative group of a local field · Local fields
• Representations and complete reducibility · Open Mathematics Courses (Representations of finite groups)
• Infinite Galois theory, Kummer theory, global class field theory and Chebotarev's theorem (course NT-CFT); Dirichlet's unit theorem (course NT-ANT); local fields and tame ramification (course NT-LOC); Hilbert polynomials and dimension theory (course AG-CA); Maschke's theorem (course RT-FIN); the Sylow theorems (core course Abstract Algebra II).