Compatible valuations and generalized Milnor 𝐾-theory

Abstract

Given a field F F and a subgroup S S of F × F^{\times } there is a minimal group S ≤ H S ≤ F × S\leq H_{S}\leq F^{\times } for which there exists an S S -compatible valuation whose units are contained in H S H_{S} . Assuming that S S has finite index in F × F^{\times } and contains ( F × ) p (F^{\times })^{p} for p p prime, we describe H S H_{S} in computable K K -theoretic terms.