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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 \({\mathbb K}_2({\mathcal O}_K)\), 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)].

MSC:

11R21 Other number fields
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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[1] Carpenter, J., Finiteness theorems for forms over global fields, Math. Zeit., 209, 153-166 (1992) · Zbl 0724.11021
[2] J. W. CASSELS andA. FRÖhlich,Algebraic Number Theory. Academic Press, 1967. · Zbl 0153.07403
[3] Conner, P. E.; Perus, R.; Szymiczek, K., Wild sets and 2-ranks of class groups, Acta Arithm., 79, 83-91 (1997) · Zbl 0880.11039
[4] Czogala, A., On reciprocity equivalence of quadratic number fields, Acta Arithm., 58, 365-387 (1991)
[5] Czogala, A., On integral Witt equivalence of algebraic number fields, Acta Math, et Inform. Univ. Ostraviensis, 4, 7-20 (1996) · Zbl 0870.11022
[6] Czogala, A.; Sladek, A., Higher degree Hubert symbol equivalence of number fields, Tatra Mountains Math. Publ., 11, 77-88 (1997) · Zbl 0978.11058
[7] Czogala, A.; Sladek, A., Higher degree Hubert symbol equivalence of number fields II, J. of Number Theory, 72, 363-376 (1998) · Zbl 0922.11096
[8] Milnor, J., Algebraic K-Theory and quadratic forms, Invent. Math., 9, 318-344 (1970) · Zbl 0199.55501
[9] J. Neukirch,Class Field Theory. Springer, 1986. · Zbl 0587.12001
[10] Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers (1990), PWN Warszawa: Springer, PWN Warszawa · Zbl 0717.11045
[11] Perlis, R.; Szymiczek, K.; Conner, P.; Litherland, R., Matching Witts with global fields, Contemp. Math., 155, 365-387 (1994) · Zbl 0807.11024
[12] Sladek, A., Hubert symbol equivalence and Milnor K-functor, Acta Math. et Inform. Univ. Ostraviensis, 6, 183-189 (1998) · Zbl 1024.11068
[13] Szymiczek, K., Witt equivalence of global fields, Commun. Alg., 19, 4, 1125-1149 (1991) · Zbl 0724.11020
[14] _, Tame Equivalence and Wild Sets.Semigroup Forum (To appear).
[15] Szymiczek, K., A characterization of tame Hilbert-symbol equivalence, Acta Math. et Inform. Univ. Ostraviensis, 6, 191-201 (1998) · Zbl 1024.11022
[16] _,Bilinear Algebra. Gordon and Breach, 1997.
[17] Täte, J., Relations betweenK_2 and Galois cohomology, Invent. Math., 36, 257-274 (1976) · Zbl 0359.12011
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