THE GROUP $ K_3$ FOR A FIELD

Abstract

This paper gives a description of the torsion and cotorsion in the Milnor groups and for an arbitrary field . The main result is that, for any natural number with , and the group is uniquely -divisible if . This theorem is a consequence of an analogue of Hilbert's Theorem 90 for relative -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 , where is the ideal of classes of even-dimensional forms in the Witt ring of the field , as well as a more complete description of the group for all global fields.