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

2 · Going further

The main problem concerns a function field $K$ of transcendence degree at least two over an algebraically closed field, with $\ell \ne \mathrm{char}(K)$. The data are $K_1^{\mathrm{M}}(K)/\ell$, $K_2^{\mathrm{M}}(K)/\ell$, and their full bilinear product. The main manuscript establishes that they determine the field’s perfect closure and its constant field. Every compatible isomorphism of these data comes from a field isomorphism up to a single scalar in ${\mathbb{F}}_{\ell }^{\times }$; in positive characteristic, the field isomorphism has only Frobenius ambiguity.

The second manuscript studies characteristic $p$ with reduction modulo $p$. For finitely generated extensions $K$ and $L$ of transcendence degree at least two over algebraically closed fields, it is shown in the manuscript that every isomorphism of their first Milnor groups modulo $p$ preserving the degree-two Steinberg relations is a nonzero scalar multiple of the map induced by a unique field isomorphism. The explicitly stated argument recovers projective lines over the subfields of $p$th powers through an elementary calculation with derivations. The repository summary states that this reconstruction recovers the original field and its named base.

The third manuscript returns to $\ell \ne \mathrm{char}(K)$ but uses the pro-$\ell$ abelian-by-central Galois datum. It is shown in that manuscript that this datum determines the perfect closure and constants, with precisely Frobenius and $\ell$-adic unit ambiguities. The abstracts of the first and third manuscripts do not detail their arguments: they chiefly identify sufficient data and the remaining ambiguities. The main result has a Lean formalization, but the supplied text does not specify formal coverage of every claim in this family.

Useful notions

• Fields and field extensions
• Characteristic and arithmetic modulo a prime
• Transcendence degree
• Groups and bilinear maps
• Basic Galois theory

What was known before / what is new

Limits and assumptions

The results require transcendence degree at least two and algebraically closed base fields; the second manuscript explicitly requires finitely generated extensions. Away from the characteristic, the stated target is the perfect closure with its constants, not generally the original field without ambiguity. The first result retains a nonzero scalar factor and, in positive characteristic, Frobenius ambiguity; the Galois result has Frobenius and ℓ-adic unit ambiguities. In equal characteristic, the second manuscript states uniqueness of the field isomorphism after accounting for the nonzero scalar factor. The main result has a Lean formalization, but the source supplies no more detailed scope or confirmation of formalization for the companions. It lists no remaining open problem and supports no conclusion outside these hypotheses.

Mini glossary

Transcendence degree

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The number of algebraically independent quantities needed to describe a field extension.

Algebraically closed field

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A field in which every nonconstant polynomial with coefficients in that field has a root.

Perfect closure

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In positive characteristic p, the extension obtained by adjoining successive pth roots of elements; in characteristic zero, the field is already perfect.

Bilinear product

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A combination rule that respects addition separately in each of its two inputs.

Frobenius

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In characteristic p, the map sending an element x to its pth power.

Keywords : Field reconstruction · Milnor K-theory · Galois data

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family