Regulators of an Infinite Family of the Simplest Quartic Function Fields

Abstract

Abstract. We explicitly nd regulators of an in nite family {L m } of the simplest quartic function elds with a parameter m in a polynomial ring F q [t], where F q is thenite eld of order q with odd characteristic. In fact, this in nite family of the simplest quartic function elds are sub elds of maximal real sub elds of cyclotomic function elds, where they have the same conductors. We obtain a lower bound on the class numbers of the family {L m } and some result on the divisibility of the divisor class numbers of cyclotomic function elds which contain {L m } as their sub elds. Furthermore, we nd an explicit criterion for the characterization of splitting types of all the primes of the rational function eld F q (t) in {L m }.