{
  "id": "open-homomorphisms-of-solvably-closed-galois-groups",
  "title": "Open homomorphisms of solvably closed Galois groups",
  "description": "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.",
  "units": [
    {
      "id": "solvably-closed-fields-and-uchidas-conjecture",
      "title": "Solvably closed fields and Uchida's conjecture",
      "reader": "solvably-closed-fields-and-uchidas-conjecture.html",
      "source": "src/solvably-closed-fields-and-uchidas-conjecture.md",
      "source_sha256": "4f766158fb43a50e2e405eaca41929f090eb38f9eb1bc6f316681a7e1aa681c2",
      "status": "author_self_checked"
    },
    {
      "id": "counting-irreducible-components",
      "title": "Counting irreducible components",
      "reader": "counting-irreducible-components.html",
      "source": "src/counting-irreducible-components.md",
      "source_sha256": "df5f25d150e664ceb9c52822d03684f9a62a46ca659a06f1fc86b5c6a2847447",
      "status": "author_self_checked"
    },
    {
      "id": "a-logarithm-rank-lemma",
      "title": "A logarithm rank lemma",
      "reader": "a-logarithm-rank-lemma.html",
      "source": "src/a-logarithm-rank-lemma.md",
      "source_sha256": "60ada2b5f6f577ad0775a9e2f66fcfdc3781ce977ed2cb246ce907e657cb5a2d",
      "status": "author_self_checked"
    },
    {
      "id": "additive-characters-and-frobenius-detection",
      "title": "Additive characters and Frobenius detection",
      "reader": "additive-characters-and-frobenius-detection.html",
      "source": "src/additive-characters-and-frobenius-detection.md",
      "source_sha256": "96cf1fc927152c4f51d6c296cd6dcaec0cf80a888aef75ae15491ed9870e1fe7",
      "status": "author_self_checked"
    },
    {
      "id": "normalizer-extension-and-frobenius-generation",
      "title": "Normalizer extension and Frobenius generation",
      "reader": "normalizer-extension-and-frobenius-generation.html",
      "source": "src/normalizer-extension-and-frobenius-generation.md",
      "source_sha256": "a3d25a4ba3cd93a8569e17ed92a5c75b6bf56b26f972a76513aea2791e5c1a72",
      "status": "author_self_checked"
    },
    {
      "id": "kummer-classes-and-s-units",
      "title": "Kummer classes and S-units",
      "reader": "kummer-classes-and-s-units.html",
      "source": "src/kummer-classes-and-s-units.md",
      "source_sha256": "fbb79535c15f4e3c1af8d6bb33b26cacb8b886e301b691690384e8198c514e67",
      "status": "author_self_checked"
    },
    {
      "id": "matching-residue-characteristics",
      "title": "Matching residue characteristics",
      "reader": "matching-residue-characteristics.html",
      "source": "src/matching-residue-characteristics.md",
      "source_sha256": "65f72d03a62313fc33dbc9dbc295e573f999c8409f2c617b85cd8390919f3a0e",
      "status": "author_self_checked"
    },
    {
      "id": "identifying-the-homomorphism",
      "title": "Identifying the homomorphism",
      "reader": "identifying-the-homomorphism.html",
      "source": "src/identifying-the-homomorphism.md",
      "source_sha256": "bce54d3e611e8b1b74ee39a429d88e34b9111324098771a6537f752423813812",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course NT-CFT, lessons Profinite groups and infinite Galois theory (Theorem 1.2), Hilbert's Theorem 90 and Kummer theory (Theorems 3.1 and 3.3), Cyclotomic polynomials and their automorphisms, The global reciprocity law (Proposition 16.1), Ray class fields, conductors and ideal reciprocity (Proposition 18.1), Global existence and the idelic class field correspondence, The Chebotarev density theorem (Theorem 22.4)",
    "course NT-ANT, lessons Decomposition of primes in extensions and Dirichlet's unit theorem (Theorems 9.2 and 9.4)",
    "course NT-LOC, lessons NT-LOC-03 (Theorem 1.1, Newton lifting), NT-LOC-05 (Proposition 4.1 and the product formula, Theorem 5.2), NT-LOC-08 (Theorem 3.1, tame inertia), NT-LOC-09 (Corollary 2.2) and NT-LOC-11 (Proposition 2.1 and Theorem 3.1)",
    "course NT-DIRL, lesson Dirichlet's theorem on primes in arithmetic progressions (Theorem 4.1)",
    "course AG-CA, lessons The Nullstellensatz and Jacobson rings (Theorem 2.2), Associated primes and primary decomposition (Theorem 3.2), Krull dimension and Noether normalization (Proposition 1.2), Graded modules and Hilbert-Samuel functions (Theorem 3.1), Dimension theory of Noetherian local rings (Theorem 2.1)",
    "course RT-FIN, lesson Representations and complete reducibility (Theorem 2.3, Maschke, and Corollary 4.1)",
    "core course Abstract Algebra II (C40): Judson, Abstract Algebra: Theory and Applications, Section 15.1 (the Sylow theorems)",
    "the named inputs (U1) Uchida's isomorphism theorem, (U2) Uchida's partial local correspondence and (H) Hoshi's criterion with Uchida's uniqueness, stated in the first lesson and used without proof"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}