Reconstruction from Milnor K-theory modulo the characteristic

Abstract — v1

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 $p$th powers by an elementary calculation with derivations.