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 $K_h$ 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.
