Quotients of Milnor \(K\)-rings, orderings, and valuations (Q997108)

Let \(F\) be a field. The Milnor \(K\)-ring \(K_*^M(F)\) of \(F\) is defined to be \((F^*)^{\otimes r}/I_r\), where \(I_r\) is the homogeneous ideal generated by all elements \(a_1\otimes \cdots \otimes a_r,\) where \(1=a_i+a_j\) for some \(1\leq i < j\leq r\) [Milnor 1970]. The author generalizes this definition to the following: Let \(S\) be a subgroup of \(F^*.\) Define the graded ring \(K_*^M(F)/ S\) to be \((F^*/ S )^{\otimes r} / I_{r, S}\) where \(I_{r, S}\) is the homogeneous ideal generated by all elements \(a_1S\otimes \cdots \otimes a_r S\) with \(1=a_i S+a_j S\) for some \(1\leq i < j\leq r\). In particular, if \(S={1}\) then \(K_*^M(F)/ S = K_*^M(F)\) and if \(S=(F^*)^m,\) then \(K_*^M(F)/ S = K_*^M(F)/ m.\) The author studies the relationship between the structure of \(K_*^M(F)/ S\) and some arithmetical properties of \(F\), especially those related to orderings and valuations.