Bounds of ideal class numbers of real quadratic function fields

Details Bounds of ideal class numbers of real quadratic function fields Bounds of ideal class numbers of real quadratic function fields

2000-04-07

arxiv.org/abs/math/0004190v1

Mathematics

Number Theory

Scientific paper

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

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