Reconstructing function fields from Milnor K-theory

Abstract

Let $F$ be a finitely generated regular field extension of transcendence degree $\ge 2$ 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 $F$ over $k$ 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 $K_{\ast }^M\left(F\right)$ determines the isomorphism class of $F$, when $k$ is algebraically closed or finite.