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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

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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