The group k3for a field

A. S. Merkur'ev, A. A. Suslin

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


This paper gives a description of the torsion and cotorsion in the Milnor groups K3M(F) and A3(F)nd= coker(K3M(F) ®A3(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 K3M(F) 2®1(F)3I(F)4where 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 K3for all global fields.

Original languageEnglish (US)
Pages (from-to)541-565
Number of pages25
JournalMathematics of the USSR - Izvestija
Issue number3
StatePublished - Jun 30 1991

ASJC Scopus subject areas

  • General Mathematics


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