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On a duality of Gras between totally positive and primary cyclotomic units. (English) Zbl 1408.11109

Summary: Let \(K\) be a real abelian field of odd degree over \(\mathbb Q\), and \(C\) the group of cyclotomic units of \(K\). We denote by \(C_+\) and \(C_0\) the totally positive and primary elements of \(C\), respectively. G. Gras [cf. Bull. Soc. Math. France 103, 177–190 (1975; Zbl 0312.12013)] found a duality between the Galois modules \(C_+/C^2\) and \(C_0/C^2\) by some ingenious calculation on cyclotomic units. We give an alternative proof using a consequence (=“Gras conjecture”) of the Iwasawa main conjecture and the standard reflection argument. We also give some related topics.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R18 Cyclotomic extensions
11R29 Class numbers, class groups, discriminants

Citations:

Zbl 0312.12013
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