Valuations and diameters of Milnor \(K\)-rings
It is convenient to begin the review by quoting the main theorem of the paper, after which we will define the necessary objects: ``Let \(F\) be a (commutative) field and let \(S\) be a subgroup of \(F^{\times}\) containing \(-1\). Suppose that the graph of \(K_*^M(F)/S\) is finite and of diameter \(\geq 4\). Then there exists a nontrivial valuation \(\nu\) on \(F\) such that \(S\) is open in the \(\nu\)-topology.'' In this way the paper provides a method for obtaining, from graph- and \(K\)-theoretic data, a suitable valuation on the field. The construction of valuations from Milnor \(K\)-theory goes back to work by \textit{B. Jacob} [J. Algebra 68, 247--267 (1981; Zbl 0457.10007)] on Pythagorean fields, while graphs appear in the following similar, non-commutative result by \textit{A. S. Rapinchuk} and \textit{Y. Segev} [Invent. Math. 144, No.~3, 571--607 (2001; Zbl 0999.16016)]: Let \(D\) be a finite-dimensional division algebra over a finitely-generated field and \(S\) a finite index, normal subgroup of \(D^{\times}\). If the diameter of the commuting graph of \(D^{\times}/S\) is \(\geq4\), then there exists a non-trivial valuation on \(D\) for which \(S\) is open (the commutating graph of a group has the non-trivial elements as vertices, with an edge between two elements if and only if they commute). Here \(K_*^M(F)/S=\bigoplus_{r=0}^\infty K_r^M(F)/S\) is the author's relative Milnor \(K\)-ring [\textit{I. Efrat}, Pac. J. Math. 226, No.~2, 259--275 (2006; Zbl 1161.19002)] which, in degree \(r\), is defined to be \((F^{\times}/S)^{\otimes r}\) modulo the relations \(a_1S\otimes\dots\otimes a_rS\) whenever \(1\in a_iS+a_jS\) for some \(i\neq j\). The associated graph is defined to have vertices for each non-trivial coset of \(F^{\times}/S\), with two vertices \(aS, bS\) connected by an edge if and only if \(\{a,b\}_S\) (\(=\) the image of \(aS\otimes bS\) in \(K_2^M(F)/S\)) is zero. Finally, the diameter of a graph is the greatest integer \(d\) (possibly \(\infty\)) for which there exist two vertices such that the shortest path between them contains exactly \(d\) edges. The proofs in the paper are elementary (though by no means obvious!) calculations, occasionally mimicking \textit{A. S. Rapinchuk} and \textit{Y. Segev} [loc. cit.] and borrowing one deep result from \textit{V. Bergelson} and \textit{D. B. Shapiro} [Proc. Am. Math. Soc. 116, No.~4, 885--896 (1992; Zbl 0784.12002)]: namely that if \(F\) is an infinite field and \(S\) a finite index subgroup of \(F^{\times}\), then \(F=S-S\). Various arithmetic examples are given in section \(6\), and section \(7\) presents an example to show that the bound of the diameter being \(\geq 4\) is strict.
scientific article; zbMATH DE number 5610598
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