IMAGINARY BICYCLIC FUNCTION FIELDS WITH THE REAL CYCLIC SUBFIELD OF CLASS NUMBER ONE

Abstract

Let <TEX>$k={\mathbb{F}}_q(T)$</TEX> and <TEX>${\mathbb{A}}={\mathbb{F}}_q[T]$</TEX>. Fix a prime divisor <TEX>${\ell}$</TEX> q-1. In this paper, we consider a <TEX>${\ell}$</TEX>-cyclic real function field <TEX>$k(\sqrt[{\ell}]P)$</TEX> as a subfield of the imaginary bicyclic function field K = <TEX>$k(\sqrt[{\ell}]P,\;(\sqrt[{\ell}]{-Q})$</TEX>, which is a composite field of <TEX>$k(\sqrt[{\ell}]P)$</TEX> wit a <TEX>${\ell}$</TEX>-cyclic totally imaginary function field <TEX>$k(\sqrt[{\ell}]{-Q})$</TEX> of class number one. und give various conditions for the class number of <TEX>$k(\sqrt[{\ell}]{P})$</TEX> to be one by using invariants of the relatively cyclic unramified extensions <TEX>$K/F_i$</TEX> over <TEX>${\ell}$</TEX>-cyclic totally imaginary function field <TEX>$F_i=k(\sqrt[{\ell}]{-P^iQ})$</TEX> for <TEX>$1{\leq}i{\leq}{\ell}-1$</TEX>.