Regulators of an Infinite Family of theSimplest Quartic Function Fields

Lee, Jungyun ; Lee, Yoonjin

Canadian journal of mathematics, Tome 69 (2017) no. 3, pp. 579-594

Résumé

We explicitly find regulators of an infinite family $\{{{L}_{m}}\}$ of the simplest quartic function fields with a parameter $m$ in a polynomial ring ${{\mathbb{F}}_{q}}\left( t \right)$ , 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 having 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 that 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 ${{\mathbb{F}}_{q}}\left( t \right)$ in $\{{{L}_{m}}\}$ .

Détail

DOI : 10.4153/CJM-2016-038-2

Mots-clés : 11R29, 11R58, regulator, function field, quartic extension, class number