Existence of integral bases for relative extensions of \(n\)-cyclic number fields
type: Item
instance of: scholarly article
title: Existence of integral bases for relative extensions of \(n\)-cyclic number fields
zbMATH Open document ID: 0871.11072
author: Xianke Zhang
author: Fuhua Xu
published in: Journal of Number Theory
publication date: 1997-10-01T00:00:00Z
review text: Sufficient conditions are given for an abelian extension \(L/K\) to have a relative integral basis when \(\text{Gal}(L/\mathbb{Q})\cong (\mathbb{Z}/q^s\mathbb{Z})^n\) and \(\text{Gal}(K/\mathbb{Q})\cong (\mathbb{Z}/q^r\mathbb{Z})^m\). Here, \(L=K_1K_2\dots K_n\) where each \(K_i/\mathbb{Q}\) is a cyclic extension of degree \(q^s\). For any prime \(p\), and an abelian extension \(F/\mathbb{Q}\), let \(e(p,F)\) denote the ramification index of \(p\) in \(F\) and \(q^{e_p}=\max_{l\leq i\leq n}(p,K_i)\). Let \([L:K]=q^M\). If \(q\) is an odd prime and \(M\geq e_p\) for all prime numbers \(p\) with \(e(p,L)\neq e(p,K)\) then \(L/K\) has a relative integral basis. For \(q=2\), a similar, but more complicated condition is given for the existence of a relative integral basis.
reviewed by: Charles J. Parry
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 955267
zbMATH Keywords: abelian extension
zbMATH Keywords: relative integral basis
MaRDI profile type: Publication
full work available at URL: jnth.1996.0131
OpenAlex ID: W2009103382
DOI: 10.1006/JNTH.1996.0131
description: scientific article; zbMATH DE number 955267