Simplest cubic fields
type: Item
instance of: scholarly article
title: Simplest cubic fields
zbMATH Open document ID: 0851.11059
DOI: 10.1007/BF02567804
author: Franz Lemmermeyer
author: Attila Pethő
published in: Manuscripta Mathematica
publication date: 1996-11-11T00:00:00Z
full work available at URL: https://eudml.org/doc/156109
review text: Let \(a\) be a natural number, \(f_a = x^3 - ax^2 - (a + 3)x - 1\), and let \(K\) be the cyclic cubic field generated by a zero \(\alpha\) of \(f_a\). Suppose that \(a^2 + 3a + 9\) is squarefree. The main result is: For all \(\gamma \in \mathbb{Z} [\alpha]\), either \(|N_\gamma |\geq 2a + 3\), or \(\gamma\) is associated to a rational integer. If \(|N \gamma |= 2a + 3\), then \(\gamma\) is associated to a conjugate of \(\alpha - 1\). Using this result, the authors construct unramified extensions \(L/K\) with \(\text{Gal} (L/K) \simeq C_2 \times C_2\) in certain cases.
Mathematics Subject Classification ID: 11R16
zbMATH DE Number: 866149
zbMATH Keywords: unramified abelian quartic extension
zbMATH Keywords: cyclic cubic field
reviewed by: Veikko Ennola
MaRDI profile type: Publication
cites work: A Device for Generating Fields of Even Class Number
cites work: Arithmetic properties of a class of cubic cyclic fields
cites work: The Simplest Cubic Fields
cites work: Class Numbers of the Simplest Cubic Fields
cites work: On certain cubic fields. IV
OpenAlex ID: W2000368716
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description: scientific article; zbMATH DE number 866149