Ideal class groups and subgroups of real quadratic function fields (Q2739080)

\textit{X. Zhang} and \textit{L. C. Washington} obtained a necessary and sufficient conditions for the ideal class group \(H(E)\) of any real quadratic number field \(F=\mathbb{Q}(\sqrt{d})\) to have a cyclic subgroup of order \(n\) [Ideal class groups and their subgroups of real quadratic fields, Sci. China, Ser. A 27, 522-528 (Chinese) (1997); English version 40, 909-916 (1997; Zbl 0907.11039) (the second author is missing there)]. In this paper, the authors intend to obtain some analogous results for real quadratic function fields by using the theory of continued fractions of algebraic functions and give some necessary and sufficient conditions for real quadratic function fields.NEWLINENEWLINENEWLINELet \(K =k(\sqrt{D})\) be a real quadratic function field over the rational function field \(k/F_q\), where \(D\) is a square-free monic polynomial with even degree, and suppose that for a nonconstant polynomial \(a\), \(A=2a+1\) is monic. Then for the following types of square-free polynomials \(D=(A^d+a)^2\pm A\), \(D=(A^d\pm(a+1))^2-A\) they show that all the ideal class groups of \(K= k(\sqrt{D})\) contain a cyclic subgroup of order \(n\) and hence that all the ideal class numbers of \(K= k(\sqrt{D})\) have a factor \(n\).