Longueurs des décompositions en produits d'éléments irréductibles dans un anneau de Dedekind. (Lengths of decompositions in products of irreducible elements in a Dedekind ring)
type: Item
instance of: scholarly article
title: Longueurs des décompositions en produits d'éléments irréductibles dans un anneau de Dedekind. (Lengths of decompositions in products of irreducible elements in a Dedekind ring)
zbMATH Open document ID: 0593.13015
DOI: 10.1016/0021-8693(86)90138-9
author: Jean-Luc Steffan
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let A be a Dedekind domain and C(A) be its group of ideal classes. It is shown that there exists a constant \(q\geq 1\) such that A satisfies \(E_ q\), namely, for any two finite families \(a_ 1,...,a_ r\) and \(b_ 1,..,b_ s\) of irreducible (extremal) elements of A, if the ideal generated by the a's is equal to the ideal generated by the b's, then 1/q\(\leq r/s\leq q\). If we define \(\ell (C(A))\) as the smallest integer m such that for each family of m elements of C(A) we can extract a subfamily whose sum is zero, then A satisfies \(E_ q\) with \(q=\ell (C(A))/2\). This bound is attained when each non-principal class of ideals of A contains at least one prime ideal of A. In \(propositions\quad 3,\) it is shown hat when C(A) is isomorphic to the direct sum of cyclic subgroups generated by the elements of \(P_ A\) which is the set of ideal classes containing at least one non-principal ideal, then A satisfies \(E_ q\) with \(q=m/2\). Here m is the largest of the orders of elements of \(P_ A\) and q is the best possible. The results above are used to get information on the number and invertibility of the roots of some families of polynomials over A and to get a lower bound for the number of ideal classes when A is the ring of integers of an imaginary quadratic field.
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 3952922
zbMATH Keywords: Dedekind domain
zbMATH Keywords: number of ideal classes
reviewed by: N. Sankaran
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90138-9
OpenAlex ID: W2011326768
cites work: Q5521594
cites work: A Characterization of Algebraic Number Fields with Class Number Two
cites work: Q1082376
cites work: Half factorial domains
description: scientific article; zbMATH DE number 3952922