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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 ${𝕂}_{2}\left({𝒪}_{K}\right)$, 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)].

##### MSC:
 11R21 Other number fields 19F15 Symbols and arithmetic ($K$-theory) 11E81 Algebraic theory of quadratic forms
##### Keywords:
Hilbert-symbol equivalence; tame kernels
##### References:
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