Quotients of Milnor K-rings, Orderings, and Valuations

We define and study the Milnor K-ring of a field F modulo a subgroup of the multiplicative group of F . We compute it in several arithmetical situations, and study the reflection of orderings and valuations in this ring. Introduction Let F be a field and let F× be its multiplicative group. The Milnor K-ring K ∗ (F ) of F is the tensor (graded) algebra of the Z-module F× modulo the homogenous ideal generated by all elements a1⊗· · ·⊗ar, where 1 = ai +aj for some 1 ≤ i < j ≤ r [Mi]. Alongside with K ∗ (F ), the quotients K M ∗ (F )/m = K M ∗ (F )/mK M ∗ (F ), where m is a positive integer, also play an important role in many arithmetical questions. In this paper we study a natural generalization of these two functors. Specifically, we consider a subgroup S of F× and define the graded ring K ∗ (F )/S to be the quotient of the tensor algebra over F×/S modulo the homogeneous ideal generated by all elements a1S ⊗ · · · ⊗ arS, where 1 ∈ aiS + ajS for some 1 ≤ i < j ≤ r. The graded rings K ∗ (F ) and K ∗ (F )/m then correspond to S = {1} and S = (F×)m, respectively. The ring-theoretic structure of K ∗ (F )/S reflects many of the main arithmetical properties of F , especially those related to orderings and valuations. We illustrate this by computing it in the following situations: ∗) The research has been supported by the Israel Science Foundation grant No. 8008/02–1 2000 Mathematics subject classification: Primary 19F99, Secondary 12J10, 12J15, 12E30