Reconstructing function fields from Milnor K-theory

TL;DR

This paper demonstrates that the Milnor K-theory of a function field over a perfect base field uniquely determines the field itself, revealing a deep connection between algebraic K-theory and field structure.

Contribution

It establishes that the Milnor K-ring encodes enough information to reconstruct the isomorphism class of the function field in a functorial manner.

Findings

Milnor K-theory determines the field up to isomorphism.

The multiplicative group modulo constants encodes algebraic dependence.

Results apply to fields over algebraically closed or finite base fields.

Abstract

Let $F$ be a finitely generated regular field extension of transcendence degree $\geq 2$ over a perfect field $k$. We show that the multiplicative group $F^\times/k^\times$ endowed with the equivalence relation induced by algebraic dependence on $k$ determines the isomorphism class of $F$ in a functorial way. As a special case of this result, we obtain that the isomorphism class of the graded Milnor $K$-ring $K^M_*(F)$ determines the isomorphism class of $F$, when $k$ is algebraically closed or finite.