Result 009, Number theory

Function-field reconstruction from Milnor K-theory and Galois data

Reconstructs function fields of transcendence degree at least two over algebraically closed constants from K1M/ℓK^{\mathrm M}_1/\ell, K2M/ℓK^{\mathrm M}_2/\ell, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.

Lean formalization Proof

The bigger picture

Why it matters

A function field is a field of rational functions, where addition and multiplication interact. These manuscripts claim that limited information about multiplication can recover such a field, or a precisely defined enlargement of it.

What changes?

For finitely generated fields over algebraically closed constants, with at least two algebraically independent variables, the inputs are the first two Milnor K-groups modulo a prime ell and their product. The first records nonzero elements modulo ell-th powers; the second records relations between pairs. Away from the characteristic, the manuscripts report recovery of the constants and perfect closure, which adds every iterated p-th root in characteristic p. Modulo the characteristic, they recover the original field and its named base.

What does that help mathematicians do?

The claims also constrain which transformations of these data are possible. Compatible isomorphisms must come from field isomorphisms, up to one nonzero scalar modulo ell. Away from the characteristic, the field isomorphism has only Frobenius ambiguity in positive characteristic; modulo the characteristic, it is unique. Thus researchers could rule out compatible transformations unrelated to field structure, making these invariants tools for comparing fields rather than merely distinguishing some examples.

Are there practical applications?

The immediate value is foundational, connecting reconstruction to Galois theory. In the same setting away from the characteristic, another manuscript reports recovery of the perfect closure and constants from pro-ell abelian-by-central Galois data: symmetries of prime-power extensions retaining a central layer of commutators. It specifies Frobenius and ell-adic unit ambiguities. This identifies precisely what field structure survives in that restricted Galois information.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory

October 5, 2026 36 pages

We prove a mod-ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The groups K1M/ℓK^{\mathrm M}_1/\ell and K2M/ℓK^{\mathrm M}_2/\ell, together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in Fℓ×\mathbb F_\ell^\times, with only Frobenius ambiguity in the field isomorphism in positive characteristic.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026,
  author = {{OpenAI}},
  title = {{Reconstruction of Function Fields from Mod-$\ell$ Milnor $K$-Theory}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf}{OAI:Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026}},
  year = {2026}
}

Reconstruction from Milnor K-theory modulo the characteristic

October 5, 2026 11 pages

Let K and L be finitely generated extensions of transcendence degree at least two over algebraically closed fields of characteristic p. We prove that every isomorphism of their first Milnor K-groups modulo p preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The proof recovers projective lines over the subfields of pth powers by an elementary calculation with derivations.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026,
  author = {{OpenAI}},
  title = {{Reconstruction from Milnor $K$-theory modulo the characteristic}},
  howpublished = {OpenAI Math Release preprint
    \href{https://github.com/openai/math/blob/main/preprints/Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026/paper.pdf}{OAI:Reconstruction-from-Milnor-K-theory-modulo-the-characteristic-October-5-2026}},
  year = {2026}
}

The Bogomolov-Pop reconstruction theorem

September 23, 2026 26 pages

We prove the Bogomolov–Pop reconstruction conjecture for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The pro-ℓ abelian-by-central datum determines the perfect closure and its constant field, with precisely the Frobenius and ℓ-adic unit ambiguities.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Bogomolov-Pop-reconstruction-theorem-September-23-2026,
  author = {{OpenAI}},
  title = {{The Bogomolov--Pop reconstruction theorem}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf}{OAI:The-Bogomolov-Pop-reconstruction-theorem-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/009.md.

Function-field reconstruction from Milnor K-theory and Galois data

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Bogomolov–Pop reconstruction conjecture concerns recovering a function field and its constants from pro-ℓ\ell abelian-by-central Galois data. The linked formalization proves injectivity of the reconstruction correspondence for function fields of transcendence degree at least two over algebraically closed fields of characteristic different from the prime ℓ\ell.

If two isomorphisms of perfect closures preserving the constant fields induce the same Galois-data isomorphism modulo ℓ\ell-adic unit scaling, then they agree modulo Frobenius. The selected statement proves this uniqueness; existence of a field isomorphism for every admissible Galois-data isomorphism is outside it.

Comparator links

Result Comparator statement
Injectivity of Bogomolov–Pop reconstruction modulo the stated ambiguities BogomolovPopInjectivity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.