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On the torsion in \(K_ 2\) of local fields. (English) Zbl 0519.12010

The author proves the following conjecture of J. Tate in [Algebraic Number Theory, Pap. Kyoto Int. Symp. 1976, 243–261 (1977; Zbl 0368.12008)] : Let \(F\) be a local field of \(\mathrm{char}\, 0\) and \(n\) the number of roots of unity in \(F\). Then the group \(K_2(F)\) is a direct sum of a cyclic group of order \(n\) and the uniquely divisible group \(nK_2(F)\). The analogous result for \(\mathrm{char}\, F\ne 0\) is due to Tate (loc. cit.).
The proof is done in two steps: First of all it is shown, that the result is true for local fields, which satisfy some special conditions on the roots of unity and on ramification. Then it is proved that any local field is contained in one of these special fields, from which the result follows via descent.

MSC:

11S70 \(K\)-theory of local fields
11S15 Ramification and extension theory
19C99 Steinberg groups and \(K_2\)
19F05 Generalized class field theory (\(K\)-theoretic aspects)

Citations:

Zbl 0368.12008
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