zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 Hilbert-symbol equivalence of two number fields K and L where 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 =2 and K, L are quadratic number fields.

Moreover, the author finds some new invariants of the tame degree Hilbert-symbol equivalence, among them the -rank of the tame kernel 𝕂 2 (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:
11R21Other number fields
19F15Symbols and arithmetic (K-theory)
11E81Algebraic theory of quadratic forms
References:
[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.
[3]P. E. Conner, R. Perus, andK. Szymiczek, Wild sets and 2-ranks of class groups.Acta Arithm. 79 (1997), 83–91.
[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.
[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.
[10]W. Narkiewicz,Elementary and Analytic Theory of Algebraic Numbers. PWN Warszawa, Springer 1990.
[11]R. Perlis, K. Szymiczek, P. Conner, andR. Litherland, Matching Witts with global fields.Contemp. Math. 155 (1994), 365–387.
[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.
[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