Cyclic cubic monogenic extensions of algebraic integers of a quadratic field
type: Item
instance of: scholarly article
title: Cyclic cubic monogenic extensions of algebraic integers of a quadratic field
zbMATH Open document ID: 0943.11046
published in: Archiv der Mathematik
publication date: 1999-07-20T00:00:00Z
review text: Let \(A_d\) denote the ring of integers of the quadratic number field \(\mathbb Q ( \sqrt d)\), \(d \neq 3\). This paper exhibits criteria for the existence of an irreducible, cubic polynomial \(P \in A_d [X]\) with cyclic Galois group and discriminant equal to \(1\). Any root of such a polynomial then generates the ring of integers of a relatively unramified, cyclic, cubic extension of \(\mathbb Q ( \sqrt d)\). The author checked that 3730 square-free integers \(d\) with \(|d|< 100 000\) satisfy these criteria, for 69 of them he found a polynomial \(P\) with the above properties.
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 1295490
zbMATH Keywords: existence of irreducible cubic polynomial
zbMATH Keywords: ring of integers
zbMATH Keywords: cyclic cubic extension
author: Jean-Daniel Therond
reviewed by: Günter Lettl
MaRDI profile type: Publication
full work available at URL: s000130050320
OpenAlex ID: W405962505
DOI: 10.1007/S000130050320
description: scientific article; zbMATH DE number 1295490