Regulators and class numbers of an infinite family of quintic function fields
Volume 185 / 2018
Abstract
We explicitly determine regulators and the system of fundamental units of an infinite family of totally real quintic function fields with a parameter h in a polynomial ring \mathbb{F}_q [t], where \mathbb{F}_q is the finite field of order q=p^r with characteristic \not=5. We use the notion of Lagrange resolvents of the generating quintic polynomials of K_h. In fact, this infinite family of quintic function fields are subfields of maximal real subfields of cyclotomic function fields, where they have the same conductors. As an application, we obtain a result on the divisibility of the divisor class numbers of maximal real subfields k(\Lambda_{P(h)})^+ of cyclotomic function fields with the same conductor P(h) as K_h. Furthermore, we obtain infinitely many irregular primes of second class f(t) \in \mathbb{F}_q[t] such that h(k(\Lambda_{f})^+) \equiv 0 ({\rm mod}\ {p^4}). Moreover, we find an explicit formula for the ideal class number of K_h and a lower bound for those numbers.