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Reconstruction from Milnor K-theory modulo the characteristic
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Abstract
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.
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Comments and discussion
- AHGT project / Alena Pirutka, On reconstruction from Milnor K-theory modulo the characteristic (Taming the Flow seminar announcement, scheduled December 18, 2026). The programme announces Pirutka's forthcoming discussion of this manuscript in the context of logarithmic differential forms and the Bloch–Gabber–Kato theorem.
Related prior literature mentioned by others
- Anna Cadoret and Alena Pirutka, Reconstructing function fields from Milnor K-theory (Algebra & Number Theory 15, 2021; linked author version August 28, 2019). Cited in the AHGT talk's references, its positive-characteristic recovery of projective lines over the subfield of pth powers provides context for the manuscript's differential intersection method.
- Spencer Bloch and Kazuya Kato, p-adic étale cohomology (Publications Mathématiques de l'IHÉS 63, 1986). The AHGT announcement cites this work when explaining the logarithmic-differential interpretation of Milnor K-theory modulo the characteristic relevant to the manuscript.
Possibly relevant literature (found by the agent, not checked)
- S. Scully, Milnor K-groups and function fields of hypersurfaces in positive characteristic (arXiv:1202.6604, 2012). Studies mod-p Milnor K-groups of function fields in characteristic p by way of Kähler differentials and the Bloch–Gabber–Kato theorem, which is close to the logarithmic-derivative pairing used in Section 2.
Digests and expositions
- AI Proved What?, The whole field from a shadow (Episode 009, October 2026). This AI-assisted video and transcript explain the three reconstruction claims, outline the mod-ℓ argument, and distinguish the Galois uniqueness formalization from the full theorem. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 009
Independent sites; their content is not reviewed here.
Cite
@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}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/009.md. The Lean statements have not been checked by people against the paper's claims.
Family 009: Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $K^{\mathrm M}_1/\ell$, $K^{\mathrm M}_2/\ell$, 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.
Other papers in this family:
- Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory · October 5, 2026
- The Bogomolov-Pop reconstruction theorem · September 23, 2026