The norm residue homomorphism of degree three

An analogue of Hilbert's Theorem 90 is proved for the Milnor groups of the fields $K_3^M$. Specifically, let $L/F$ be a quadratic extension, and let be the generator of the Galois group. Then the sequence
$$ K_3^M(L)\stackrel{1-\sigma}{\longrightarrow}K_3^M(L)\stackrel{N}{\longrightarrow}K_3^M(F) $$

is exact. As a corollary one can prove bijectivity of the norm residue homomorphism of degree three:
$$ K_3^M(F)/2^nK_3^M(F)\to H^3(F,\mu_{2^n}^{\otimes 3}). $$
Finally, the 2-primary torsion in $K_3^M(F)$ is described: if the field $F$ contains a primitive $2^n$th root of unity $\xi$, then the $2^n$-torsion subgroup of $K_3^M(F)$ is $\{\xi\}\cdot K_2(F)$.