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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 \(\Delta_m\).

MSC:

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

Citations:

Zbl 1099.11029
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References:

[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
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