Indivisibility of divisor class numbers of Kummer extensions over the rational function field
type: Item
instance of: scholarly article
title: Indivisibility of divisor class numbers of Kummer extensions over the rational function field
zbMATH Open document ID: 1415.11163
author: Jinjoo Yoo
author: Yoonjin Lee
published in: Journal of Number Theory
publication date: 2018-09-24T00:00:00Z
review text: Let \(K\) be a Kummer extension over the rational function field \(k={\mathbb F}_q(T)\) of degree \(\ell\), where \({\mathbb F}_q\) is the finite field of order \(q\) and \(\ell\) is a prime divisor of \(q-1\). In Theorems 1.1 and 1.2 of the paper under review the authors obtain results regarding the indivisibility of the divisor class number \(h_K\) of \(K\) by \(\ell\) in two cases, respectively. In the first case they assume that the infinite prime of \(k\) is ramified in \(K\) and prove that \(h_K\) is not divisible by \(\ell\) if and only if there exists exactly one finite prime of \(k\) which is ramified in \(K\). Moreover, when \(h_K\) is not divisible by \(\ell\), then \(h_K\equiv 1\pmod \ell\). Equivalently, the divisor class number \(h_K\) of \(K\) is divisible by \(\ell\) if and only if there exists at least two finite primes of \(k\) which are ramified in \(K\). In the second case it is assumed that the infinite prime of \(k\) is unramified in \(K\). In this case the authors prove that \(h_K\) is not divisible by \(\ell\) if and only if the number of finite primes of \(k\) which are ramified in \(K\) is at most two and one of the following three cases holds: \begin{itemize}\item[(i)] \(K\) is an unramified extension over \(k\); \item[(ii)] If there is only one finite prime of \(k\) which is ramified in \(K\), then the genus of \(K\) is divisible by \(\ell\)\text; \item[(iii)] If there are exactly two finite primes \(P_1\) and \(P_2\) of \(k\), of degree \(d_1\) and \(d_2\), respectively, which are ramified in \(K\), then \(d_i\not\equiv 0 \pmod \ell\) for \(i=\text{1}\) or \(i=\text{2}\). \end{itemize} In all three cases (i), (ii), and (iii), we have \(h_K\equiv 1 \pmod \ell\). Using the above indivisibility criterion, the authors obtain in Theorem 1.3 an infinite family of maximal real cyclotomic function fields whose divisor class numbers are divisible by \(\ell\) where \(\ell\) is a prime divisor of \(q-1\). More precisely, for an irreducible monic po…
reviewed by: James E. Carter
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 6943098
zbMATH Keywords: Kummer extension
zbMATH Keywords: class number
zbMATH Keywords: cyclotomic function field
zbMATH Keywords: global function field
Wikidata QID: Q114157313
MaRDI profile type: Publication
full work available at URL: j.jnt.2018.04.016
OpenAlex ID: W2807526534
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DOI: 10.1016/J.JNT.2018.04.016
description: scientific article; zbMATH DE number 6943098