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Regulators and class numbers of an infinite family of quintic function fields

Volume 185 / 2018

Jungyun Lee, Yoonjin Lee Acta Arithmetica 185 (2018), 107-125 MSC: 11R29, 11R58. DOI: 10.4064/aa8626-1-2018 Published online: 17 July 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.

Authors

  • Jungyun LeeInstitute of Mathematical Sciences
    Ewha Womans University
    Seoul 03760, S. Korea
    e-mail
  • Yoonjin LeeDepartment of Mathematics
    Ewha Womans University
    Seoul 03760, S. Korea
    e-mail

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