BULANT, Michal. On Washington group of circular units of some composita of quadratic fields. Math. Slovaca, Bratislava, 2005, vol. 55, No 1, p. 39-46. ISSN 0139-9918.
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Basic information
Original name On Washington group of circular units of some composita of quadratic fields
Name in Czech Washingtonova grupa kruhových jednotek některých komposit kvadratických těles
Authors BULANT, Michal (203 Czech Republic, guarantor).
Edition Math. Slovaca, Bratislava, 2005, 0139-9918.
Other information
Original language English
Type of outcome Article in a journal
Field of Study 10101 Pure mathematics
Country of publisher Czech Republic
Confidentiality degree is not subject to a state or trade secret
RIV identification code RIV/00216224:14310/05:00012374
Organization unit Faculty of Science
Keywords in English circular units; abelian field; class number
Tags abelian field, circular units, class number
Changed by Changed by: Mgr. Michal Bulant, Ph.D., učo 2759. Changed: 18/4/2005 10:30.
Abstract
Circular units emerge in many occasions in algebraic number theory as they have tight connection (first discovered by E. Kummer) to the class group of the respective number field. For example, E. Kummer has shown that in the case of cyclotomic field with prime conductor the index of the group of circular units in the full group of units is equal to the class number of the maximal real subfield of that field. His result was later generalized so we are now able to obtain information about class groups by the study of circular units. In contrast to the case of cyclotomic field it is not clear how to define the group of circular units of a general abelian number field K. In the literature there eventually turned up several possible definitions of a group of circular units. One of these definitions (which appeared in the Washington's book Introduction to cyclotomic fields) constructs the group of circular units to be as large as possible - it considers all circular units of the respective cyclotomic superfield which are lying already in the field K. This definition has some nice properties but also serious difficulties: generally we do not know neither explicit generators of the group nor the index of the group in the full group of units. In this paper we present results about this index for some classes of abelian fields - namely for composita of quadratic fields satisfying an additional condition obtained by the study of the relation between Washington group of circular units and the well-known Sinnott s group of circular units.
Abstract (in Czech)
Kruhové jednotky se často objevují v moderní algebraické teorii čísel zejména díky tomu, že jsou v úzkém vztahu z grupou tříd ideálů příslučného číselného tělesa. Vzhledem k tomu, že není jasné jak rozšířit obvyklou definici grupy kruhových jednotek kruhového tělesa na obecný případ abelovského tělesa, existuje více definic. Jednou z nich je tzv. Washingtonova grupa kruhových jednotek, u které však obecně neznáme ani generátory, ani index v grupě všech jednotek. V článku jsou prezentovány výsledky, popisující tento index pro jistou třídu abelovských těles - komposita kvadratických těles, splňující další podmínku na vztah větvících se prvočísel.
Links
GA201/04/0381, research and development projectName: Metody teorie čísel
Investor: Czech Science Foundation, Standard Projects
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