Indivisibility of divisor class numbers of Kummer extensions over the rational function field

JOURNAL OF NUMBER THEORY

卷:192

Article

Lee, Yoonjin¹  Yoo, Jinjoo¹ 

[1] Ewha Womans Univ, Dept Math, 52 Ewhayeodae Gil, Seoul 03760, South Korea

关键词: Kummer extension;    Class number;    Cyclotomic function field;    Global function field;   

DOI  :  10.1016/j.jnt.2018.04.016

来源: Elsevier

【 摘 要 】

We find a complete criterion for a Kummer extension K over the rational function field k = F-q(T) of degree l to have indivisibility of its divisor class number h(K) by l, where F-q is the finite field of order q and l is a prime divisor of q - 1. More importantly, when h(K) is not divisible by l, we have h(K) (math) 1 (mod l). In fact, the indivisibility of h(K) bye depends on the number of finite primes ramified in K/k and whether or not the infinite prime of k is unramified in K. Using this criterion, we explicitly construct an infinite family of the maximal real cyclotomic function fields whose divisor class numbers are divisible by l. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free