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On Washington group of circular units of some composita of quadratic fields. (English) Zbl 1108.11082
Summary: 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 book L. C. Washington, Introduction to Cyclotomic Fields (2nd ed.). Graduate Texts in Mathematics. 83. New York, NY: Springer (1997; Zbl 0966.11047)) 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. Methods of this paper use and slightly extend the approach appeared in R. Kučera [Circular units and class groups of abelian fields. In: Théorie des nombres et applications. Comptes Rendus de la conférence internationale Maroc-Québec (Mai 2003), Ann. Sci. Math. Qué. 28, No. 1–2, 121–136 (2004; Zbl 1103.11031)].

MSC:
11R27 Units and factorization
11R20 Other abelian and metabelian extensions
11R29 Class numbers, class groups, discriminants
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References:
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