Uchida’s conjecture for open homomorphisms of Galois groups

The explanation

Start with the field made by adding √2 and √3 to the fractions. A symmetry of this field is a way to reshuffle it that respects addition and multiplication and fixes every fraction. Each one sends √2 to √2 or −√2, and √3 to √3 or −√3. That gives 2 × 2 = 4 symmetries. Together they form the field’s Galois group (its group of symmetries). The smaller field built from √2 alone has just 2 symmetries.

The smaller field sits inside the bigger one. So every symmetry of the big field restricts to a symmetry of the small one. This gives a map from the group of 4 onto the group of 2, with two symmetries landing on each. The field map goes one way, small into big. The group map goes the other way, big onto small. That reversal is the pattern Uchida’s conjecture is about.

The question is whether every such group map comes from fields. The paper studies very large fields called solvably closed: fields with no further abelian extension (no extension whose symmetries all commute)Source 1. The field of all algebraic numbers is one example. Neukirch began reconstructing number fields from their Galois groups in 1969Source 2. Uchida proved in 1979 that two solvably closed fields with matching groups are themselves matchedSource 3. That is a statement about matches, not about general maps.

In 1981 Uchida conjectured that every continuous homomorphism with open image comes from a field embeddingSource 4. A homomorphism is a map between groups that respects how symmetries combine. Open image means, roughly, that the image is a large part of the target group. Uchida proved the conjecture when the source field is the rationals, and proved uniqueness in generalSource 5. In 2025 Hoshi showed exactly what was missing. Such a map comes from fields if and only if it preserves the full cyclotomic characterSource 6. That character records how a symmetry acts on roots of unity, the solutions of zⁿ = 1.

The paper claims the conjecture in fullSource 7. Every continuous open homomorphism between these Galois groups comes from a unique field embedding in the opposite directionSource 7. What the map sends to the identity, its kernel, is unrestrictedSource 8. The paper derives the needed arithmetic match from openness itselfSource 8. The √2 and √3 example only shows the pattern. Its fields are far too small for the theorem, which needs solvably closed fields.

The obstacle is to get Hoshi’s cyclotomic condition from local information about primes. Uchida’s local method pairs primes on the two sides using an auxiliary prime ℓ. But which primes get paired, and which are missed, depends on ℓSource 9. The paper bounds the number of missed primes without depending on ℓSource 10. It also claims a tool of wider use. A certain kind of character on the Galois group is fixed by its values on any infinite set of primes of a special kindSource 11.

The result is about solvably closed fields only. It does not say that smaller fields, such as the rationals on their own, are recovered from maps of their groups.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. The proof also uses a special case of a transcendence theorem, with its own proof given in the paperSource 12. No specialist review has been published.

What it doesn't show. This does not cover fields that still have abelian extensions, such as the rational numbers on their own.