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Regulators of an Infinite Family of the Simplest Quartic Function Fields

Published:2016-12-06
Printed: Jun 2017
• Jungyun Lee,
Institute of Mathematical Sciences, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Republic of Korea
• Yoonjin Lee,
Institute of Mathematical Sciences, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Republic of Korea
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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 $\{L_m\}$.
 Keywords: regulator, function field, quartic extension, class number
 MSC Classifications: 11R29 - Class numbers, class groups, discriminants 11R58 - Arithmetic theory of algebraic function fields [See also 14-XX]

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