Relative Class Number One Problem of Real Quadratic Fields and Continued Fraction of $\sqrt{m}$ with Period $6$
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For a positive square-free integer $m$, let $K=\mathbb{Q}(\sqrt{m})$ be a real quadratic field. The relative class number $H_d(f)$ of $K$ of discriminant $d$is the ratio of class numbers $\mathcal{O}_K$ and $\mathcal{O}_f$, where $\mathcal{O}_K$ is the ring of integers of $K$ and $\mathcal{O}_f$is the order of conductor $f$ given by $\mathbb{Z}+f\mathcal{O}_K$. In 1856, Dirichlet showed that for certain $m$ there exists an infinite number of $f$ such thatthe relative class number $H_d(f)$ is one. But it remained open as to whether there exists such an $f$ for each $m$. In this paper, we give a result forexistence of real quadratic field $\mathbb{Q}(\sqrt{m})$ with relative class number one where the period of continued fraction expansion of $\sqrt{m}$ is $6$.
Abstract
For a positive square-free integer $m$, let $K=\mathbb{Q}(\sqrt{m})$ be a real quadratic field. The relative class number $H_d(f)$ of $K$ of discriminant $d$is the ratio of class numbers $\mathcal{O}_K$ and $\mathcal{O}_f$, where $\mathcal{O}_K$ is the ring of integers of $K$ and $\mathcal{O}_f$is the order of conductor $f$ given by $\mathbb{Z}+f\mathcal{O}_K$. In 1856, Dirichlet showed that for certain $m$ there exists an infinite number of $f$ such thatthe relative class number $H_d(f)$ is one. But it remained open as to whether there exists such an $f$ for each $m$. In this paper, we give a result forexistence of real quadratic field $\mathbb{Q}(\sqrt{m})$ with relative class number one where the period of continued fraction expansion of $\sqrt{m}$ is $6$.
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