Existence of relative integral basis over quadratic fields and Pólya property
type: Item
instance of: scholarly article
title: Existence of relative integral basis over quadratic fields and Pólya property
zbMATH Open document ID: 1488.11164
author: Abbas Maarefparvar
published in: Acta Mathematica Hungarica
publication date: 2021-08-03T00:00:00Z
review text: For an algebraic number field \(K\), \textit{H. B. Mann} [Proc. Amer. Math. Soc. 9, 167--172 (1958; Zbl 0081.26602)] gave a criterion for a quadratic extension \(L\) of \(K\) to have an integral basis over \(K\). Specifically, \(L\) has an integral basis over \(K\) if and only if \(L=K(\sqrt{\alpha})\) for some \(\alpha\) for which \((\alpha)\) is the discriminant of \(L\) over \(K\). From this, he showed that every quadratic extension \(L\) of \(K\) has an integral basis over \(K\) if and only if \(K\) has class number 1 (see also \textit{W. Narkiewicz} [Elementary and analytic theory of algebraic numbers, 2nd ed., Springer-Verlag (1990; Zbl 0717.11045)], Corollary to Proposition 7.16, p. 403).\par \textit{R. H. Bird} and \textit{C. J. Parry} [Pac. J. Math. 66, 29--36 (1976; Zbl 0356.12003)] later considered the special case \(K=\mathbb{Q}(\sqrt{m})\) and \(L=\mathbb{Q}(\sqrt{m},\sqrt{n})\) in detail, where \(m,n\) are distinct squarefree rational integers. In this case, explicit and concrete conditions under which \(L\) has an integral basis over \(K\) were given, and an explicit integral basis was produced whenever one exists. In the cases where \(K\) is imaginary or \(K\) is real and has a unit of norm \(-1\), the conditions involve only rational congruences on \(m\) and \(n\). In the remaining case that \(K\) is real and has only units of norm \(+1\), the conditions sometimes also involve the fundamental unit.\par In the paper under review, the following theorem is proven. Let \(K=\mathbb{Q}(\sqrt{m})\) be a quadratic field. Then every biquadratic field \(L=\mathbb{Q}(\sqrt{m},\sqrt{n})\), for every squarefree integer \(n\) distinct from \(m\), has a relative integral basis over \(K\) if and only if \(K\) is a Pólya field, or equivalently that all strongly ambiguous ideal classes of \(K\) are trivial.\par For a general number field \(F\), the Pólya property can be characterized in various ways. For example, let \(\mathcal{O}_F\) denote the ring of integers of \…
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 7377610
zbMATH Keywords: quadratic field
zbMATH Keywords: biquadratic field
zbMATH Keywords: relative integral basis
zbMATH Keywords: Pólya field
MaRDI profile type: Publication
full work available at URL: s10474-021-01158-2
OpenAlex ID: W3168623550
cites work: From Pólya fields to Pólya groups. I: Galois extensions
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cites work: On Integral Bases
cites work: Q4821147
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cites work: Integer valued polynomials over a number field
DOI: 10.1007/S10474-021-01158-2
description: scientific article; zbMATH DE number 7377610