Lower bound for ideal class numbers of real quadratic function fields (Q2739079)

\textit{K. Feng} and \textit{W. Hu} obtained a lower bound of ideal class numbers for real quadratic function fields [Proc. Am. Math. Soc. 127, 1301-1307 (1999; Zbl 0924.11089)]. In this paper, the authors intend to generalize this result by using the theory of continued fractions of algebraic functions and obtain some results as follows: Let \(K=k(\sqrt{D})\) be a real quadratic function field over the rational function field \(k/F_q\) and suppose that \(P\) is an irreducible polynomial of \(k\) splitting in \(K\) with \(\text{deg }P\leq d-1\), \(2d = \text{deg }D\). Then for any of six types of \(D = A^2+c\), \((A^m+a)^2+A\) and so on, \(h(K)\geq\text{deg }A/\text{deg }P\) holds, where \(h(K)\) is the ideal class number of \(K\). Consequently, they show moreover that there exist six classes of real quadratic function fields with ideal class number greater than one.