Proceedings of the Japan Academy, Series A, Mathematical Sciences

On a certain invariant for real quadratic fields

Seok-Min Lee and Takashi Ono

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Abstract

Let $K = \mathbf{Q}(\sqrt{m})$ be a real quadratic field, $\mathcal{O}_K$ its ring of integers and $G = \operatorname{Gal}(K/\mathbf{Q})$. 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 $M_c/P_c \approx \mathbf{Z}/\Delta_m \mathbf{Z}$ where $\Delta_m = 1$ or 2 and we will determine $\Delta_m$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 79, Number 8 (2003), 119-122.

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116443712

Digital Object Identifier
doi:10.3792/pjaa.79.119

Mathematical Reviews number (MathSciNet)
MR2013090

Zentralblatt MATH identifier
1161.11392

Subjects
Primary: 11R11: Quadratic extensions 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
Secondary: 11A07: Congruences; primitive roots; residue systems

Keywords
Real quadratic field fundamental unit parity continued fractions

Citation

Lee, Seok-Min; Ono, Takashi. On a certain invariant for real quadratic fields. Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 8, 119--122. doi:10.3792/pjaa.79.119. https://projecteuclid.org/euclid.pja/1116443712


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References

  • Ono, T.: A Note on Poincaré sums for finite groups. Proc. Japan Acad., 79A, 95–97 (2003).
  • Stark, H. M.: An Introduction to Number Theory. The MIT Press, Cambridge, Massachusetts-London, England (1978).