## 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 F_{q} [t], where 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 F_{q}(t) in {L_{m}}.

Original language | English |
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Pages (from-to) | 579-594 |

Number of pages | 16 |

Journal | Canadian Journal of Mathematics |

Volume | 69 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2017 |

### Bibliographical note

Publisher Copyright:© Canadian Mathematical Society 2016.

## Keywords

- Class number
- Function field
- Quartic extension
- Regulator