## Abstract

This paper gives a description of the torsion and cotorsion in the Milnor groups K_{3}^{M}(F) and A_{3}(F)_{nd}= coker(K_{3}^{M}(F) ®A_{3}(F)) for an arbitrary field F. The main result is that, for any natural number n with (char/, n) = 1, (Formula Presented) and the group K3(F)nd is uniquely /-divisible if/= char F. This theorem is a consequence of an analogue of Hilbert’s Theorem 90 for relative K2-groups of extensions of semilocal principal ideal domains. Among consequences of the main result we obtain an affirmative solution of the Milnor conjecture on the bijectivity of the homomorphism K_{3}^{M}(F) 2®1(F)^{3}I(F)^{4}where I(F) is the ideal of classes of even-dimensional forms in the Witt ring of the field F, as well as a more complete description of the group K_{3}for all global fields.

Original language | English (US) |
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Pages (from-to) | 541-565 |

Number of pages | 25 |

Journal | Mathematics of the USSR - Izvestija |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - Jun 30 1991 |

## ASJC Scopus subject areas

- Mathematics(all)