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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 ${\Bbb K}_2({\cal O}_K)$, thus generalizing one of the results proved by {\it P. E. Conner, R. Perlis} and {\it K. Szymiczek} [Acta Arith. 79, 83-91 (1997; Zbl 0880.11039)].

11R21Other number fields
19F15Symbols and arithmetic ($K$-theory)
11E81Algebraic theory of quadratic forms
Full Text: DOI
[1] J. Carpenter, Finiteness theorems for forms over global fields.Math. Zeit. 209 (1992), 153--166. · Zbl 0736.11024 · doi:10.1007/BF02570827
[2] J. W. CASSELS andA. FRÖhlich,Algebraic Number Theory. Academic Press, 1967. · Zbl 0153.07403
[3] P. E. Conner, R. Perus, andK. Szymiczek, Wild sets and 2-ranks of class groups.Acta Arithm. 79 (1997), 83--91. · Zbl 0880.11039
[4] A. Czogala, On reciprocity equivalence of quadratic number fields.Acta Arithm. 58 (1991),365--387.
[5] --,On integral Witt equivalence of algebraic number fields.Acta Math, et Inform. Univ. Ostraviensis 4 (1996), 7--20. · Zbl 0870.11022
[6] A. Czogala andA. Sladek, Higher degree Hubert symbol equivalence of number fields.Tatra Mountains Math. Publ. 11 (1997), 77--88.
[7] --, Higher degree Hubert symbol equivalence of number fields II.J. of Number Theory 72 (1998), 363--376. · Zbl 0922.11096 · doi:10.1006/jnth.1998.2266
[8] J. Milnor, Algebraic K-Theory and quadratic forms.Invent. Math. 9 (1970), 318--344. · Zbl 0199.55501 · doi:10.1007/BF01425486
[9] J. Neukirch,Class Field Theory. Springer, 1986. · Zbl 0587.12001
[10] W. Narkiewicz,Elementary and Analytic Theory of Algebraic Numbers. PWN Warszawa, Springer 1990. · Zbl 0717.11045
[11] R. Perlis, K. Szymiczek, P. Conner, andR. Litherland, Matching Witts with global fields.Contemp. Math. 155 (1994), 365--387. · Zbl 0807.11024
[12] A. Sladek, Hubert symbol equivalence and Milnor K-functor.Acta Math. et Inform. Univ. Ostraviensis 6 (1998), 183--189.
[13] K. Szymiczek, Witt equivalence of global fields.Commun. Alg. 19(4) (1991), 1125- 1149. · Zbl 0724.11020 · doi:10.1080/00927879108824194
[14] _, Tame Equivalence and Wild Sets.Semigroup Forum (To appear).
[15] --, A characterization of tame Hilbert-symbol equivalence.Acta Math. et Inform. Univ. Ostraviensis 6 (1998), 191--201. · Zbl 1024.11022
[16] _,Bilinear Algebra. Gordon and Breach, 1997.
[17] J. Täte, Relations betweenK 2 and Galois cohomology.Invent. Math. 36 (1976), 257--274. · Zbl 0359.12011 · doi:10.1007/BF01390012