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Circular units and class groups of abelian fields. (English) Zbl 1103.11031
This paper gives a nice survey of circular units in abelian number fields. First, the various groups of circular units (of conductor level, those of Sinnott, of cyclic subfields and those of Washington) are put into relation to each other (e.g. in the field \(\mathbb Q (\sqrt{13}, \sqrt{17})\) these are 4 different groups). Their special features are pointed out, and several new results on the indices between these groups are presented. Also their importance for (algebraic) class number formulas for real abelian fields is discussed. Finally it is recalled how annihilators of units modulo circular units can produce annihilators of the class group, by surveying results of F. Thaine, K. Rubin, V. Kolyvagin, and C. Greither and the author [Acta Arith. 112, No. 2, 177–198 (2004; Zbl 1065.11089)].

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