On Washington group of circular units of some composita of quadratic fields

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.

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

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 project: Name: Metody teorie čísel

Investor: Czech Science Foundation, Methods of number theory