East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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RELATIVE CLASS NUMBER ONE PROBLEM OF REAL QUADRATIC FIELDS AND CONTINUED FRACTION OF $\sqrt{m}$ WITH PERIOD 6

  • Lee, Jun Ho (Department of Mathematics Education, Mokpo National University)
  • Received : 2021.09.23
  • Accepted : 2021.09.30
  • Published : 2021.09.30
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Abstract

Abstract. For a positive square-free integer m, let K = ℚ($\sqrt{m}$) be a real quadratic field. The relative class number Hd(f) of K of discriminant d is the ratio of class numbers 𝒪K and 𝒪f, where 𝒪K is the ring of integers of K and 𝒪f is the order of conductor f given by ℤ + f𝒪K. In 1856, Dirichlet showed that for certain m there exists an infinite number of f such that the relative class number Hd(f) is one. But it remained open as to whether there exists such an f for each m. In this paper, we give a result for existence of real quadratic field ℚ($\sqrt{m}$) with relative class number one where the period of continued fraction expansion of $\sqrt{m}$ is 6.

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Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(the Ministry of Education and MSIT)(2017R1D1A1B03033560, 2020R1F1A1A01069118).

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