The norm residue homomorphism of degree~three

A. S. Merkur'ev ; A. A. Suslin

Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 349-367

Résumé

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 $\{\xi\}\cdot K_2(F)$.