Groups remember fields?

0:00Symmetries and primes

The polynomial x³ − 2 has three roots: the real cube root of two, and two complex ones. A symmetry of the algebraic numbers that fixes the rationals must permute these roots, and here all six permutations occur. Collect the symmetries of all algebraic numbers at once, and you get the absolute Galois group of the rationals: a single, enormous group. Primes see these symmetries. Modulo a prime p other than 2 and 3, x³ − 2 splits into three linear factors, or a linear times a quadratic, or doesn’t factor at all, depending on a symmetry attached to p, its Frobenius element. Among the primes from five to 10,000, the three cases occur 16%, 50% and 33% of the time, close to 1/6, 1/2 and 1/3, the shares of the three kinds of permutations. Roots of unity carry a special record. The Frobenius element at p raises every root of unity of order prime to p to the p-th power. This is the cyclotomic character, and it remembers p.

1:03Fields from groups

In the late 1960s and 1970s, Neukirch and Uchida proved a striking theorem: a number field is determined by its absolute Galois group. Isomorphic groups come from isomorphic fields, and every isomorphism of the groups comes from exactly one isomorphism of the fields. For p-adic fields, the analogous statement is false. In 1981, Uchida asked for more. A field embedding induces a map between Galois groups, in the opposite direction. Is every continuous homomorphism with open image, between Galois groups of solvably closed extensions of number fields, induced by an embedding? He proved this when the first group is a Galois group over the rationals, proved uniqueness, and matched primes to primes through their inertia groups. In a paper published in 2025, Hoshi showed that the answer is yes exactly when the homomorphism preserves the cyclotomic character.

2:03The claim

Result 031 in OpenAI’s math repository claims the full conjecture. The preprint, 23 pages, dated October 5, states: every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields comes from a unique field embedding, with no condition on the kernel. Solvably closed means the field has no nontrivial abelian extensions, like the field of all algebraic numbers. There’s no Lean proof.

2:35How the proof goes

By Hoshi’s criterion, the task is to show the homomorphism preserves the cyclotomic character; in effect, it must match primes lying over the same p. Uchida’s correspondence pairs primes through inertia, using an auxiliary prime ℓ, and misses only finitely many target primes. But which ones it misses could depend on ℓ. The paper relies on two key tools. First, an additive ℓ-adic character, one with values in the ℓ-adic numbers under addition, is determined by its values at any infinite set of primes of degree one. The key rank estimate is a special case of the Waldschmidt–Masser theorem, proved here with Laurent’s interpolation determinants. Second, refining a Kummer argument of Uchida, it bounds the number of missed primes by a number that doesn’t depend on ℓ. The paper first reduces to maps onto the Galois group of the maximal solvable extension of the rationals. For such a map, it shows that it preserves the cyclotomic character after taking logarithms, for infinitely many ℓ; that corresponding primes can be forced to share their residue characteristic; and, comparing Frobenius values, that it is ordinary restriction up to a single conjugation. That gives the full cyclotomic compatibility, and Hoshi’s theorem supplies the field embedding.

3:56Where it stands

This is a preprint, not peer reviewed, with no Lean proof. It builds on the theorems of Uchida and Hoshi. If the proof holds, then for these Galois groups of number fields, every open homomorphism, and not just every isomorphism, comes from a field embedding.