The group k₃for a field

This paper gives a description of the torsion and cotorsion in the Milnor groups K₃^M(F) and A₃(F)_nd= coker(K₃^M(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 K₃^M(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.