Higher degree tame Hilbert-symbol equivalence of number fields
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 \textit{P. E. Conner, R. Perlis} and \textit{K. Szymiczek} [Acta Arith. 79, 83-91 (1997; Zbl 0880.11039)].
scientific article; zbMATH DE number 1416697
scientific article