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Math-2803-A1 · revision r1 · on Math-2803

OpenAI claim summary for Math-2803

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 031; 2026-10-05).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Open homomorphisms of solvably closed global Galois groups
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Open homomorphisms of solvably closed global Galois groups

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 031; 2026-10-05), the asserted result is that every continuous open homomorphism between the stated solvably closed number-field Galois groups arises contravariantly from a unique equivariant field embedding, with no additional kernel restriction.

Mathematical statement of the claimed result

Manuscript 1.
Let $F_1,F_2$ be number fields and $E_i/F_i$ possibly infinite Galois extensions, with each $E_i$ solvably closed, meaning it has no nontrivial abelian algebraic extension. Every continuous open homomorphism $\alpha:\operatorname{Gal}(E_1/F_1)\to\operatorname{Gal}(E_2/F_2)$ is claimed induced by exactly one field embedding $j:E_2\hookrightarrow E_1$, characterized by \[ g\circ j=j\circ\alpha(g)\quad(g\in\operatorname{Gal}(E_1/F_1)). \] In particular $j(F_2)\subseteq F_1$; finite extensions and surjectivity of $\alpha$ are not required.

Outline of the argument

For manuscript 1 (family 031; 2026-10-05), power-residue separation matches local characteristics and bounds missed primes. Regular-module Kummer extensions supply enough characters to detect Frobenius images, while averaging and corestriction turn the partial local correspondence into the required field map.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 031; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources