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Torsion in $$K_ 2$$ of fields. (English) Zbl 0635.12015
The author derives many new and fundamental results concerning the algebraic $$K$$-theory of fields. In “$$K$$-cohomology of Severi-Brauer varieties and the norm residue homomorphism” by A. S. Merkurjev and the author [cf. Math. USSR, Izv. 21, 307–340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011–1046 (1982; Zbl 0525.18008)] it was shown that if $$F$$ is a field containing an $$n$$-th root of unity, $$\xi$$, then every $$n$$-torsion element in $$K_ 2(F)$$ is of the form $$\{\xi, z\}$$ for some $$z\in F^*$$. In this paper the kernel of the map $$\{\xi,-\}: F^*\to{}_nK_ 2(F)$$ is computed, where $${}_nA=\{a\in A\mid na=0\}$$. In addition a $$K_ 2$$-reduced norm is constructed for every finite central simple algebra. Furthermore, it is shown that $$\text{Ker}(K_ 2\text{Sp}(F)\to K_ 2(F))$$ is isomorphic to $$I^3W(F)$$, the cube of the augmentation ideal in the Witt ring of $$F$$.
The paper contains several more (too many to list here) important results concerning $$K_ 2(F)$$.
Reviewer: V.P.Snaith

##### MSC:
 11R70 $$K$$-theory of global fields 12G05 Galois cohomology 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 19C30 $$K_2$$ and the Brauer group 19G12 Witt groups of rings
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