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The indecomposable \(K_ 3\) of fields. (English) Zbl 0636.18004

The author announces the following result: Theorem (Hilbert’s Theorem 90 for relative \(K_ 2)\). Let S be a semi-local p. i. d. containing a field, k, and containing an \(\ell\)-th root of unity, \(\ell\) a prime. Let J be a Jacobson radical of S and \(\alpha\) a unit in S. Set \(S^{\alpha}=S[X]/(X^{\ell}-\alpha)\) and let \(\sigma\) generate \(Gal(S^{\alpha}/S)\). Then \[ K_ 2(S^{\alpha},J^{\alpha})\to^{1- \sigma}K_ 2(S^{\alpha},J^{\alpha})\to^{norm}K\quad_ 2(S,J) \] is exact, where \(J^{\alpha}=JS^{\alpha}.\)
From this result the author derives an isomorphism between the torsion of \(K_ 3(E)^{ind}=K_ 3(E)/K_ 3^{Mi\ln or}(E)\) and \(H\) \(0_{et}(E;\mu_{\infty}^{\otimes 2})\) and calculates \(K_ 3(E;{\mathbb{Z}}/n)^{ind}\) in most cases. A. S. Merkurjev and A. A. Suslin have also obtained these results by similar methods.
Reviewer: V.Snaith

MSC:

18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
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