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 $k={ \bf F}_q(T)$. 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)\geq { deg}F /{ deg} P;$ \hspace{ 0.1cm} if $D=(SG)^2+cS, $ then $ h(D)\geq { deg}S / { deg} P; $ if $D=(A^m+a)^2+A, $ then $ h(D)\geq { deg}A / { deg} P, $ where $P$ is irreducible polynomial splitting in $K, c\in { \bf F}_q$ 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
Categories: math.NT