Regulators of an infinite family of the simplest quartic function fields

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}.