Reconstructing function fields from Milnor K-theory

Let F be a finitely generated regular field extension of transcendence degree ≥ 2 over a perfect field k. We show that the multiplicative group F ×/ k× 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 M (F) determines the isomorphism class of F, when k is algebraically closed or finite.