1 · The essentials
The manuscript asks whether a map between the symmetries of two fields must come from a map between the fields themselves. Here, fields are systems of numbers in which addition, subtraction, multiplication, and division by nonzero numbers are possible. For the particular fields and symmetry maps covered by its assumptions, the manuscript shows that the answer is yes. The corresponding map of fields goes in the opposite direction and is uniquely determined. The result allows the field extensions to be infinite and does not require two additional restrictions discussed in the source. These claims concern Uchida’s conjecture, under the precise assumptions explained below. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine trying to recover a connection between two objects using only a connection between their patterns of symmetry. The manuscript says that, for the mathematical objects and maps it covers, there is exactly one matching connection between the objects—but its arrow points the other way.
Why it matters
The result links maps of Galois groups to maps of the fields they describe, rather than treating these as unrelated kinds of information. Its scope includes possibly infinite extensions and requires neither a restriction on the kernel nor a separate cyclotomic-compatibility assumption. The manuscript presents this as a resolution of Uchida’s conjecture on open homomorphisms.
2 · Going further
The precise setting consists of solvably closed Galois extensions of number fields, which may be infinite. Given a continuous open homomorphism between their Galois groups, the manuscript shows that it is induced by a unique equivariant field embedding in the opposite direction. Thus the statement concerns not just the existence of an underlying field map, but also its uniqueness and compatibility with the group actions.
The source identifies this statement as Uchida’s conjecture on open homomorphisms of Galois groups. It explicitly removes any restriction on the kernel and any additional compatibility hypothesis; the repository summary specifically mentions that no separate cyclotomic-compatibility assumption is needed. Continuity, openness, and the specified class of field extensions remain essential parts of the stated scope.
The supplied abstract states the result but does not describe the argument or its main construction, so no proof strategy can be extracted from this source. Only one manuscript, “Open Homomorphisms of Global Solvably Closed Galois Groups,” is supplied; there are no companion manuscripts here whose relationship can be explained. The main result has not been formalized in Lean or formally verified.
Useful notions
- Fields and field embeddings
- Number fields and Galois extensions
- Galois groups and their actions
- Continuous and open homomorphisms
- Solvably closed extensions
What was known before / what is new
Before
The source identifies the question as Uchida’s conjecture on open homomorphisms of Galois groups. It supplies no account of earlier partial results or of the conjecture’s history.
New
The manuscript shows that every continuous open homomorphism in the stated setting comes from a unique equivariant field embedding in the reverse direction. The statement includes possibly infinite solvably closed Galois extensions of number fields. No restriction on the kernel or separate cyclotomic-compatibility assumption is required.
Limits and assumptions
The statement applies to continuous open homomorphisms between Galois groups of solvably closed Galois extensions of number fields, not to arbitrary groups, fields, or maps. The absence of kernel restrictions and additional compatibility assumptions does not remove these underlying hypotheses. The supplied abstract gives no argument details and identifies no remaining open questions outside this scope. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Galois group —
- The group of symmetries of a Galois extension that leave its base field fixed.
- Field embedding —
- An injective map between fields that preserves their arithmetic operations.
- Open homomorphism —
- A group homomorphism that sends open sets to open sets.
- Kernel —
- The subgroup of elements sent to the identity by a group homomorphism.
- Equivariant —
- Compatible with the group actions involved.
Keywords : Uchida’s conjecture · Galois groups · Field embeddings
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Open Homomorphisms of Global Solvably Closed Galois Groups
05/10/2026We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required.