Regulators of an Infinite Family of the Simplest Quartic Function Fields

摘要

We explicitly find regulators of an infinite family {L-m} of the simplest quartic function fields with a parameter m in a polynomial ring F-q [t], where F-q is the finite field of order q with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields having 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 fields that contain {L-m} as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field F-q(t) in {L-m}.