Geodesic
The norm residue homomorphism of degree~three
Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 349-367.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\{\xi\}\cdot K_2(F)$. 
@article{IM2_1991_36_2_a6,
     author = {A. S. Merkur'ev and A. A. Suslin},
     title = {The norm residue homomorphism of degree~three},
     journal = {Izvestiya. Mathematics },
     pages = {349--367},
     publisher = {mathdoc},
     volume = {36},
     number = {2},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a6/}
}
TY  - JOUR
AU  - A. S. Merkur'ev
AU  - A. A. Suslin
TI  - The norm residue homomorphism of degree~three
JO  - Izvestiya. Mathematics 
PY  - 1991
SP  - 349
EP  - 367
VL  - 36
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a6/
LA  - en
ID  - IM2_1991_36_2_a6
ER  - 
%0 Journal Article
%A A. S. Merkur'ev
%A A. A. Suslin
%T The norm residue homomorphism of degree~three
%J Izvestiya. Mathematics 
%D 1991
%P 349-367
%V 36
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a6/
%G en
%F IM2_1991_36_2_a6
A. S. Merkur'ev; A. A. Suslin. The norm residue homomorphism of degree~three. Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 349-367. http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a6/

[1] Merkurev A. S., “O simvole normennogo vycheta stepeni dva”, DAN SCCP, 261:3 (1981), 542–547 | MR

[2] Merkurev A. S., “Gruppa $SK_2$ dlya algebr kvaternionov”, Izv. AN CCCP. Ser. matem., 52:2 (1988), 310–335 | MR

[3] Merkurev A. S., Suslin A. A., “$K$-kogomologii mnogoobrazii Severi–Brauera i gomomorfizm normennogo vycheta”, Izv. AN SSSR. Ser. matem., 46:5 (1982), 1011–1046 | MR

[4] Merkurjev A. S., Suslin A. A., On the $K_3$ of a field, Preprint E-1-87, LOMI, 1987, 23 pp.

[5] Suslin A. A., “Algebraicheskaya $K$-teoriya i gomomorfizm normennogo vycheta”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 25, 1984, 115–209 | MR

[6] Khartskhorn R., Algebraicheskaya geometriya, Mir, M., 1981 | MR | Zbl

[7] Arason J. K., “Cohomologische Invarianten Quadratischer Formen”, J. Algebra, 36 (1975), 448–491 | DOI | MR | Zbl

[8] Bass H., Tate I., “The Milnor ring of a global field”, Lecture Notes in Math., 342, 1972, 349–446 | MR

[9] Elman R., Lam T.-Y., “Pfister forms and $K$-theory of fields”, J. Algebra, 23 (1972), 181–213 | DOI | MR | Zbl

[10] Lam T.-Y., The algebraic theory of quadratic forms, Reading. Mass., Benjamin, 1973 | MR | Zbl

[11] Merkurjev A. S., Suslin A. A., On the norm residue homomorphism of degree 3, Preprint E-9-86, LOMI, 1986

[12] Milnor I., “Algebraic $K$-theory and quadratic forms”, Invent. Math., 9:4 (1970), 318–344 | DOI | MR | Zbl

[13] Quillen D., “Higher $K$-theory”, Lecture Notes in Math., 341, 1973, 85–147 | MR | Zbl

[14] Rehmann U., Stuhler U., “On $K_2$ of finite dimensional division algebras over arithmetic fields”, Invent. Math., 50 (1978), 45–90 | DOI | MR

[15] Sherman C., “Some theorems on the $K$-theory of coherent sheaves”, Comm. Algebra, 7:14 (1979), 1489–1508 | DOI | MR | Zbl

[16] Swan R., “$K$-theory of quadric hypersurfaces”, Ann. Math., 122:1 (1985), 113–154 | DOI | MR

[17] Tate J., “Relations between $K_2$ and Galois cohomology”, Invent. Math., 36 (1976), 257–274 | DOI | MR | Zbl