## Abstract

Given a field F and a subgroup S of F^{x} containing -1, we define a graph on F^{x}/S associated with the relative Milnor K-ring K_{*}^{M}(F)/S. We prove that if the diameter of this graph is at least 4, then there exists a valuation v on F such that S is v-open. This is done by adopting to our setting a construction in a noncommutative setting due to Rapinchuk, Segev and Seitz. We study the behavior of the diameter under important K-theoretic constructions, and relate it to the elementary type conjecture. Finally, we provide an example showing that the above bound 4 is sharp.

Original language | English |
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Pages (from-to) | 75-92 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 172 |

Issue number | 1 |

DOIs | |

State | Published - 1 Aug 2009 |

## ASJC Scopus subject areas

- Mathematics (all)

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