Bounds of ideal class numbers of real quadratic function fields

Abstract

The theory of continued fractions of functions $\sqrt{D}$ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt{D})$ over $k={\mathbf{F}}_q(T)$. For five series of real quadratic function fields $K$, the bounds of $h(D)$ are given more explicitly, e.g., if $D=F^2+c,$ \mbox{}\hspace{0.1cm} then $h(D)\ge degF/degP;$ \hspace{0.1cm} if $D=(SG{)}^2+cS,$ then $h(D)\ge degS/degP;$ if $D=(A^m+a{)}^2+A,$ then $h(D)\ge degA/degP,$ where $P$ is irreducible polynomial splitting in $K,c\in {\mathbf{F}}_q$ is any constant. In addition, six types of quadratic function fields are found to have ideal class numbers bounded and bigger than one. {\bf keywords:} quadratic function field, ideal class number, continued fractions of functions