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http://dx.doi.org/10.7858/eamj.2021.040

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)
Publication Information
East Asian mathematical journal / v.37, no.5, 2021 , pp. 613-617 More about this Journal
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.
Keywords
relative class number; fundamental units;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
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