Real quadratic function fields of minimal type (Q2868486)

The authors obtain an analog of a result of \textit{F. Kawamoto} and \textit{K. Tomita} [Osaka J. Math. 46, No. 4, 949--993 (2009; Zbl 1247.11140)], to wit: There are exactly 6 class number one real quadratic function fields \(K = \mathbb F_p(x)(\sqrt{D})\) when all odd prime \(p\) are considered, such that \(D \in \mathbb F_p[x]\) is monic square-free of even degree polynomial satisfying \(\deg D > \deg u_D\), where the fundamental unit of \(K\) is of the form \(t_d + u_D \sqrt{D}\). The proof is by explicit continued fraction manipulations, combined with known class number formulas and tabulated class numbers and regulators.