Uchida’s conjecture: recovering field embeddings from Galois maps

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Family 031 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Galois theory translates field extensions into groups of symmetries. A surprising converse says that sufficiently rich Galois groups can determine the underlying number fields. Uchida’s homomorphism conjecture asks for more: can a map between such groups determine a field embedding?

The manuscript claims that every continuous homomorphism with open image between the specified global Galois groups comes from a unique field embedding in the opposite direction. No injectivity condition on the homomorphism, restriction on its kernel or extra arithmetic compatibility assumption is imposed.

The reversal of direction is familiar from restriction. A symmetry of a larger field can be restricted to a smaller field, giving a group map going the other way from the field inclusion. The difficult assertion is that an abstract group map with the stated topological property must arise from actual field arithmetic.

2. Background and essential definitions

A number field is a finite extension of $\mathbb Q$. For a Galois extension $E/F$, the group $\operatorname{Gal}(E/F)$ consists of automorphisms of $E$ fixing $F$. If the extension is infinite, this is a profinite group: an inverse limit of finite Galois groups, carrying the topology in which agreement on larger finite subextensions means closer proximity.

Open subgroups of a profinite group have finite index. In this manuscript, an open homomorphism means a continuous homomorphism with open image. Its image therefore occupies a finite-index part of the target, but the map need not be injective or surjective onto the entire target.

An algebraic extension of a number field is solvably closed if it has no nontrivial abelian algebraic extension. An abelian extension has a commutative Galois group. Algebraic closures and maximal prosolvable extensions are examples. A finite group is solvable if repeated passage to commutator subgroups eventually reaches the identity; a maximal prosolvable extension collects finite Galois extensions with solvable groups.

“Solvably closed” does not mean “algebraically closed.” It rules out further abelian extensions, while allowing the possibility of extensions with more complicated symmetry groups. This makes the theorem apply to a wider class than absolute Galois groups alone.

What it means for an embedding to induce a map

Suppose $G_i=\operatorname{Gal}(E_i/F_i)$, and let $\alpha:G_1\to G_2$. A field embedding $j:E_2\hookrightarrow E_1$ induces $\alpha$ if

$$g\circ j=j\circ\alpha(g) \qquad\text{for every }g\in G_1.$$

Both sides take an element of $E_2$ into $E_1$. This equation says that the two symmetry actions agree after transporting elements through $j$. It also implies $j(F_2)\subseteq F_1$.

The cyclotomic character records roots of unity

Every solvably closed field in this setting contains all roots of unity. An automorphism acts on an $n$th root of unity by raising it to an exponent invertible modulo $n$. Recording these compatible exponents gives the cyclotomic character

$$\chi_{E/F}:\operatorname{Gal}(E/F)\longrightarrow\widehat{\mathbb Z}^{\times},$$

where $\widehat{\mathbb Z}^{\times}$ is the inverse limit of the groups $(\mathbb Z/n\mathbb Z)^\times$. Any field embedding preserves this action. The key issue is whether an arbitrary open group homomorphism must preserve it automatically.

3. Development history and previous partial results

Neukirch began the reconstruction of number fields from their Galois groups through work on normal number fields and decomposition groups. Uchida, Iwasawa and Ikeda contributed to the passage to arbitrary number fields and to prescribed group isomorphisms. Uchida subsequently established the isomorphism theorem for arbitrary solvably closed Galois extensions of number fields.

An isomorphism carries all group information in both directions. A general homomorphism can collapse a large kernel, so the reconstruction problem is more demanding. In his 1981 paper, Uchida formulated the open-homomorphism conjecture, proved it when the source number field is $\mathbb Q$, proved uniqueness in general and established existence under a suitable local condition on decomposition groups.

Decomposition groups are subgroups recording how a chosen prime behaves in an extension. Their inertia subgroups describe ramification. Uchida’s local analysis associates source primes with target primes whenever a chosen auxiliary-prime part of inertia survives under the map. For each fixed odd auxiliary prime $\ell$, this partial correspondence reaches all but finitely many target primes. However, its domain and exceptional set may depend on $\ell$.

Hoshi later gave an exact criterion: an open homomorphism in this setting is induced by a field embedding if and only if it preserves the full cyclotomic character. This reduced the existence problem to an arithmetic compatibility condition. Further work of Hoshi reformulated the unrestricted conjecture through radicals and the normality of fields fixed by kernels.

Related results for finite solvable quotients include work of Saïdi–Tamagawa and Mao–Saïdi. These explore how much field information remains after restricting the allowed complexity of Galois groups, sometimes retaining cyclotomic compatibility as a hypothesis.

The new manuscript claims to derive the necessary full compatibility from openness alone. Its final reconstruction step uses Hoshi’s criterion; the proposed advance is removing the additional condition, rather than replacing all the earlier reconstruction theory.

4. What the new manuscripts claim

Let $F_1,F_2$ be number fields and $E_i/F_i$ possibly infinite Galois extensions, with each $E_i$ solvably closed. For every continuous open homomorphism

$$\alpha:\operatorname{Gal}(E_1/F_1) \longrightarrow\operatorname{Gal}(E_2/F_2),$$

the theorem claims a unique equivariant field embedding

$$j:E_2\hookrightarrow E_1.$$

Uniqueness refers to the specified homomorphism $\alpha$ and the equivariance equation above. If one changes identifications of algebraic closures or conjugates group maps, the corresponding formulation changes accordingly; the statement is not merely that some unspecified field embedding exists.

The theorem has no additional kernel restriction. It does not assume preservation of decomposition groups, inertia or cyclotomic characters in advance. Those arithmetic constraints are recovered within the argument.

An intermediate theorem is especially useful to state. Let $\mathbb Q^{\mathrm{sol}}$ be the maximal prosolvable extension of $\mathbb Q$, and put

$$P=\operatorname{Gal}(\mathbb Q^{\mathrm{sol}}/\mathbb Q).$$

For any number field $F$, every continuous open map from its absolute Galois group $G_F$ to $P$ is claimed to be ordinary restriction up to conjugation in $P$. The manuscript uses this identification to prove

$$\chi_{E_2/F_2}\circ\alpha=\chi_{E_1/F_1}$$

as full profinite characters, not just after applying a logarithm at one prime. Hoshi’s theorem then supplies existence, and Uchida’s earlier uniqueness result completes the conclusion.

5. Why the advance matters

Importance: 4/5, provisionally. The claim settles a major reconstruction problem in anabelian number theory, the study of how arithmetic and geometry are encoded in fundamental or Galois groups. It strengthens recognition of fields from group isomorphisms to recognition of field embeddings from a broad class of group maps.

This is a form of rigidity. A homomorphism is initially described only through group operations and topology, but the theorem says openness forces it to respect enough arithmetic to arise from a field map. In particular, the apparent freedom to alter the action on roots of unity disappears.

The result has a specific global setting: solvably closed Galois extensions of number fields. The rating reflects both its centrality within that setting and the fact that it does not classify arbitrary homomorphisms for arbitrary fields. The claim’s significance is assessed here independently of proof verification.

6. The main idea

Detect maps through additive characters

An additive $\ell$-adic character is a continuous homomorphism from a Galois group to the additive group of $\mathbb Q_\ell$. It turns a complicated noncommutative object into numerical data that can be compared at Frobenius elements.

A finite prime has absolute residue degree one if its residue field is $\mathbb F_p$ for a rational prime $p$. The manuscript uses a detection statement: an additive $\ell$-adic character vanishing at infinitely many such prime Frobenius elements is zero. Consequently, agreement of two characters on such an infinite set forces agreement everywhere. The paper also develops versions involving Frobenius averages and transfer maps to handle other residue degrees.

The input behind detection is a logarithmic-rank theorem from $\ell$-adic transcendence theory. Prime-ideal generators with suitably distinct supports have multiplicatively independent coordinates; their $\ell$-adic logarithms then provide enough independent directions to determine a linear form. The manuscript works with the actual image of the unit logarithms, without imposing a separate rank conjecture on that image.

Improve a partial local correspondence

Uchida’s construction supplies many corresponding primes for each auxiliary $\ell$, but its exceptional sets can move as $\ell$ changes. A collection of individually cofinite correspondences is not automatically one coherent arithmetic correspondence.

The manuscript first derives logarithmic cyclotomic compatibility using enlarged finite quotients and additive character tests. It then revisits the missing target primes through Kummer theory, which describes radical extensions by classes of elements modulo powers.

Each missed prime supplies a Kummer class. After removing a finite character twist, these classes lie in one character component of an $S$-unit representation. An $S$-unit is an element allowed to have nonzero valuations only at a fixed finite set $S$ of places. The multiplicity of the relevant character is bounded in terms of the fixed source field, even though the surrounding extension can grow with $\ell$.

This gives a bound on the number of missed target primes independent of $\ell$. Combined with an additional sign-character argument, it allows auxiliary primes to be chosen so that enough source and target primes have the same underlying rational residue characteristic.

Recover one conjugation for every finite quotient

Once residue characteristics align, suitably split rational primes compare the additive-character pullbacks from the candidate map with those from ordinary restriction. Infinite sets of degree-one primes force the pullbacks to agree.

A faithful action on a suitable character space then identifies the two group maps modulo each finite quotient, up to a conjugation. “Faithful” means that no nonidentity group element acts trivially, so matching all these actions detects the group element itself.

Initially the conjugating element might depend on the quotient. Compactness of the profinite group combines the compatible nonempty sets of choices into one conjugation working for every quotient. This is what upgrades many finite identifications to an identification of the whole map to $P$.

Every solvably closed field contains $\mathbb Q^{\mathrm{sol}}$, so the original homomorphism can be composed with restriction to $P$. The intermediate theorem identifies that composite. Roots of unity already lie in the solvable extension, and their action is unchanged by conjugation, yielding the full cyclotomic equality needed for reconstruction.

7. Scope, limitations, and open questions

Continuity and open image are essential hypotheses of the stated theorem. It does not classify arbitrary abstract homomorphisms, maps with nonopen image or Galois groups over unrelated classes of fields.

The fields $E_i$ must be solvably closed and Galois over number fields $F_i$. The result should not be read as a blanket theorem for every finite solvable quotient, every function field or every local field. Those settings have their own reconstruction questions.

The endpoint requires full cyclotomic compatibility. Equality of $\ell$-adic logarithms alone can lose finite torsion information, which is why the later identification of group maps is part of the argument rather than an optional refinement.

The theorem is also an existence-and-uniqueness statement, not an efficient algorithm for recovering a field embedding from a finite description of an infinite profinite group. Quantitative or computational reconstruction would require additional information beyond the claimed abstract rigidity.

8. Sources and further reading

Uchida’s Homomorphisms of Galois groups of solvably closed Galois extensions is the primary source for the conjecture, uniqueness and earlier local results. His isomorphism theorem supplies an important predecessor.

Hoshi’s cyclotomic-compatible homomorphism theorem gives the criterion used at the endpoint. Dasgupta’s Ranks of matrices of logarithms of algebraic numbers. I explains the transcendence-theoretic rank results behind the character tests. M1 below develops the uniform local comparison and its application to the unrestricted open-homomorphism claim.

Source manuscripts

• M1. OpenAI, Open Homomorphisms of Global Solvably Closed Galois Groups (October 5, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.