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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
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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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