Fitting ideals of class groups of real fields with prime power conductor
type: Item
instance of: scholarly article
title: Fitting ideals of class groups of real fields with prime power conductor
zbMATH Open document ID: 0926.11085
author: Pietro Cornacchia
author: Cornelius Greither
published in: Journal of Number Theory
publication date: 1999-11-25T00:00:00Z
review text: We quote the authors' words: ``In this paper we prove a structure result on the ideal class group and on the narrow ideal class group of totally real fields with prime power conductor. For any number field \(K\) we denote its ideal class group by \(\text{Cl}_K\), its narrow ideal class group by \(\text{Cl}_K^\infty\), its unit group by \(O_{\mathcal K}^*\) and the group of totally positive units by \(O_{{\mathcal K},+}^*\). Let \(l\) be a prime number, let \(n\in \mathbb{N}\) and suppose that \(K\) has conductor \(l^n\), that is to say \(K\) is contained in the field \(\mathbb{Q} (\zeta_{l^n})\) obtained by adjoining to \(\mathbb{Q}\) a primitive \(l^n\)th root of unity \(\zeta_{l^n}\) and \(n\) is minimal. We denote by \(G\) the Galois group \(\text{Gal} (K/\mathbb{Q})\). We define the group of cyclotomic units \(\text{Cyc}_K\) of \(K\) as \[ \text{Cyc}_K= O_{\mathcal K}^*\cap (1-\zeta_{l^n})^{\mathbb{Z}[ \text{Gal} (\mathbb{Q}(\zeta_{l^n})/ \mathbb{Q})]}. \] It is known that \(\text{Cyc}_L\) has finite index in \(O_{\mathcal K}^*\) and that the index \((O_{\mathcal K}^*: \text{Cyc}_K)\) is equal to the order of \(\text{Cl}_K\), but the two Galois modules \(\text{Cl}_K\) and \(O_{\mathcal K}^*/ \text{Cyc}_K\) are not isomorphic in general, not even as abelian groups. In this paper we prove that their Fitting ideals \(\text{Fit}_{\mathbb{Z}[G]} (\text{Cl}_K)\) and \(\text{Fit}_{\mathbb{Z}[G]} (O_{\mathcal K}^*/ \text{Cyc}_K)\) are equal. We also prove a result about the \(\mathbb{Z}[G]\)-Fitting ideal of the narrow ideal class group \(\text{Cl}_K^\infty\).
reviewed by: K. Lakkis
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 1247931
zbMATH Keywords: ideal class group
zbMATH Keywords: narrow ideal class group
zbMATH Keywords: totally real fields
zbMATH Keywords: prime power conductor
zbMATH Keywords: group of cyclotomic units
zbMATH Keywords: Fitting ideals
MaRDI profile type: Publication
full work available at URL: jnth.1998.2300
OpenAlex ID: W2063101481
cites work: Q3760711
cites work: Q5532660
cites work: Anderson's module for cyclotomic fields of prime conductor
cites work: Class groups of abelian fields, and the main conjecture
cites work: The structure of some minus class groups, and Chinburg's third conjecture for abelian fields
cites work: Some cases of Brumer's conjecture for abelian CM extensions of totally real fields
cites work: Q4855224
cites work: On formulae for the class number of real Abelian fields
cites work: Class fields of abelian extensions of \(\mathbb Q\)
cites work: Global units and ideal class groups
cites work: Q5750966
cites work: On the Stickelberger ideal and the circular units of an abelian field
cites work: Q5690489
DOI: 10.1006/JNTH.1998.2300
description: scientific article; zbMATH DE number 1247931