On a certain invariant for real quadratic fields. (English) Zbl 1161.11392

Summary: This article is a continuation and completion of the second author’s paper [Proc. Japan Acad., Ser. A 79, No. 4, 95–97 (2003; Zbl 1099.11029)]. Let \(K = \mathbb Q(\sqrt{m})\) be a real quadratic field, \(\mathcal O_K\) its ring of integers and \(G = \text{Gal}(K/\mathbb Q)\). For \(\gamma \in H^1(G, \mathcal O_K^{\times})\), we associate a module \(M_c/P_c\) for \(\gamma = [c]\). It is known that \(M_c/P_c \approx \mathbb Z/\Delta_m \mathbb Z\) where \(\Delta_m = 1\) or 2 and we determine \(\Delta_m\).


11R11 Quadratic extensions
11F75 Cohomology of arithmetic groups
11R27 Units and factorization


Zbl 1099.11029
Full Text: DOI


[1] Ono, T.: A Note on Poincaré sums for finite groups. Proc. Japan Acad., 79A , 95-97 (2003). · Zbl 1099.11029 · doi:10.3792/pjaa.79.95
[2] Stark, H. M.: An Introduction to Number Theory. The MIT Press, Cambridge, Massachusetts-London, England (1978). · Zbl 0412.03033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.