The group $K_3$ for a field

This paper gives a description of the torsion and cotorsion in the Milnor groups $K_3^M(F)$ and $K_3(F)_{nd}=\operatorname{coker}(K_3^M(F)\to K_3(F))$ for an arbitrary field $F$. The main result is that, for any natural number $n$ with $(\operatorname{char}F,n)=1$, $_nK_3(F)_{nd}=H^0(F,\mu_n^{\otimes 2})$, $K_3(F)_{nd}/n=\operatorname{ker}(H^1(F,\mu_n^{\otimes 2})\to K_2(F))$ and the group $K_3(F)_{nd}$ is uniquely $l$-divisible if $l=\operatorname{char}F$. This theorem is a consequence of an analogue of Hilbert's Theorem 90 for relative $K_2$-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 $K_3^M(F)/2\to I(F)^3/I(F)^4$, 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.