Conductor ideals of orders in algebraic number fields
type: Item
instance of: scholarly article
title: Conductor ideals of orders in algebraic number fields
zbMATH Open document ID: 1312.11084
author: Günter Lettl
author: Chanwit Prabpayak
published in: Archiv der Mathematik
publication date: 2014-10-10T00:00:00Z
review text: Let \(K\) be a number field, \(\mathcal O_K\) its ring of algebraic integers and let \(\mathcal O\) be an order of \(K\). Let us recall that the conductor \(\mathfrak f\) of \(\mathcal O\) is defined by \(\mathfrak f=\{x\in K:\;x\mathcal O_K\subseteq \mathcal O\}\). In the paper under review the authors give a new version of the proofs of results from \textit{Ph. Furtwängler} [Anzeiger Wien 56, 75 (1919; JFM 47.0146.04)] about necessary and sufficient conditions for an ideal of \(\mathcal O_K\) to be the conductor of some order of \(K\). More concretely, first of all it is shown that it is enough to study the case in which the norm of \(\mathfrak f\) is the power of a rational prime. Under this assumption, \(\mathfrak f=\mathfrak p_1^{k_1}\cdots\mathfrak p_g^{k_g}\) is shown to be a conductor if and only if for every \(1\leq i\leq q\) we have: if \(\mathfrak p_i\) is of absolute degree \(1\) and \(k_i\equiv 1\pmod{e_i}\), then there exists \(j\neq i\) such that \(k_j>\frac{k_i-1}{e_i}e_j\), where \(e_i\) is the ramification index of \(\mathfrak p_i\) over \(\mathbb Q\).
reviewed by: Alessandro Cobbe
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 6353785
zbMATH Keywords: conductor ideal
zbMATH Keywords: number field
zbMATH Keywords: ring of integers
zbMATH Keywords: orders
MaRDI profile type: Publication
full work available at URL: s00013-014-0655-z
OpenAlex ID: W2074146782
cites work: Q3448510
DOI: 10.1007/S00013-014-0655-Z
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description: scientific article; zbMATH DE number 6353785