Higher degree tame Hilbert-symbol equivalence of number fields. (English) Zbl 0968.11038

The main aim of the paper is to give necessary and sufficient conditions for the tame degree \(\ell\) Hilbert-symbol equivalence of two number fields \(K\) and \(L\) where \(\ell\) is an odd prime. The conditions are expressed in terms of the classical invariants, and are similar to those given in the author’s paper [Acta Arith. 58, 29-46 (1991; Zbl 0733.11012)] for the case where \(\ell=2\) and \(K\), \(L\) are quadratic number fields.
Moreover, the author finds some new invariants of the tame degree \(\ell\) Hilbert-symbol equivalence, among them the \(\ell\)-rank of the tame kernel \({\mathbb K}_2({\mathcal O}_K)\), thus generalizing one of the results proved by P. E. Conner, R. Perlis and K. Szymiczek [Acta Arith. 79, 83-91 (1997; Zbl 0880.11039)].


11R21 Other number fields
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
11E81 Algebraic theory of quadratic forms; Witt groups and rings
Full Text: DOI


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