Abstract view
Regulators of an Infinite Family of the Simplest Quartic Function Fields


Published:20161206
Printed: Jun 2017
Jungyun Lee,
Institute of Mathematical Sciences, Ewha Womans University, 52, Ewhayeodaegil, Seodaemungu, Seoul 03760, Republic of Korea
Yoonjin Lee,
Institute of Mathematical Sciences, Ewha Womans University, 52, Ewhayeodaegil, Seodaemungu, Seoul 03760, Republic of Korea
Abstract
We explicitly find regulators of an infinite family $\{L_m\}$
of the simplest quartic function fields
with a parameter $m$ in a polynomial ring $\mathbb{F}_q [t]$, where
$\mathbb{F}_q$
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,
where they have 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 which
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 $\mathbb{F}_q (t)$ in $\{L_m\}$.