The prime at infinity and the rank of the class group in global function fields
type: Item
instance of: scholarly article
title: The prime at infinity and the rank of the class group in global function fields
zbMATH Open document ID: 1089.11064
author: Allison M. Pacelli
published in: Journal of Number Theory
publication date: 2006-03-24T00:00:00Z
review text: Let \({\mathbb F}\) be the finite field of \(q\) elements and \(k={\mathbb F}(T)\). For any finite extension \(K\) of \(k\) let \(Cl_ K\) be the ideal class group of the integral closure \({\mathfrak O} _ K\) of \({\mathbb F}[T]\) in \(K\). The author proved in [J. Number Theory 106, No. 1, 26--49 (2004; Zbl 1100.11039)] that for any integers \(m\) and \(n\) not divisible by the characteristic there are infinitely many function fields of fixed degree \(m\) with class number divisible by \(n\). \textit{Y. Lee} and the author [Proc. Am. Math. Soc. 133, No. 10, 2883--2889 (2005; Zbl 1100.11037)] proved that for any relatively prime integers \(m\) and \(n\), not divisible by the characteristic and with some other restrictions, there exist infinitely many function fields \(K\) of degree \(m\) over \(k\) such that the infinite prime in \(k\) is inert in \(K\) and \(Cl_ K\) contains a subgroup isomorphic to \(({\mathbb Z}/n {\mathbb Z})^ {m-1}\). The objective of the paper under review is to generalize the above results. The main theorem is that if \(m\) and \(n\) are any positive integers larger than one, not both even, and not divisible by the characteristic of \(k\), and if \(h\) is an integer with \(2 \leq h \leq m-1\), then there exist infinitely many function fields \(K\) of degree \(m\) over \(k\) such that the infinite prime in \(k\) splits into exactly \(h\) prime divisors of \(K\), one with ramification index \(m-h+1\), the rest unramified, all with relative degree one and \(Cl_ K\) contains a subgroup isomorphic to \(({\mathbb Z}/ n{\mathbb Z}) ^ {m-h}\). The main tools used in this paper are the same as the ones used in the papers cited above and an application of Newton's polygon.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 5015054
zbMATH Keywords: Class number
zbMATH Keywords: class group
zbMATH Keywords: ideal class group
zbMATH Keywords: congruence
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: j.jnt.2005.04.014
OpenAlex ID: W2032989116
cites work: Q3313949
cites work: Class Number Divisibility in Real Quadratic Function Fields
cites work: Corrigendum: On the class groups of pure function fields
cites work: Class groups of imaginary function fields: The inert case
cites work: On ideal groups of algebraic number fields.
cites work: Abelian subgroups of any order in class groups of global function fields
cites work: Q2776773
cites work: On unramified Galois extensions of quadratic number fields
DOI: 10.1016/J.JNT.2005.04.014
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description: scientific article; zbMATH DE number 5015054