A family of cyclic cubic polynomials whose roots are systems of fundamental units.
type: Item
instance of: scholarly article
title: A family of cyclic cubic polynomials whose roots are systems of fundamental units.
zbMATH Open document ID: 1034.11060
DOI: 10.1016/S0022-314X(03)00085-4
author: Yasuhiro Kishi
published in: Journal of Number Theory
publication date: 2003-08-20T00:00:00Z
review text: Let \(K=\mathbb Q(\xi)\) be a cyclic cubic field. It is easy to see that \(K\) contains an element \(\theta_u\) such that \(\text{Irr}(\theta_u,\mathbb Q)\) is of the form \(x^3-ux^2-(u+3)x-1\). The family \(\{\mathbb Q(\theta_n):n\in\mathbb Z\}\) is the well-known family of simplest cubic fields which has been largely investigated. It follows that \(K\) has an automorphism \(\sigma\) such that \(\sigma(\theta_u)=M_0(\theta_u)\), where \(M_0\) is the fractional linear transformation with matrix \(\begin{pmatrix} 0&-1\\1&1\end{pmatrix}\). Take \(M_2=\begin{pmatrix} -1&0\\a&-n+a\end{pmatrix}\) for some integers \(a,n\), \(M_1=M_2M_0M_2^{-1}\), and \(\xi=M_2(\theta_u)\). Then \(\sigma(\xi)=M_1(\xi)\), and the coefficients of \(\text{Irr}(\xi,\mathbb Q)\) are integral polynomials of \(n\) for exactly four values of \(a\), namely \(a=1,-n+1,n^2+n+1,-n^3+1\). For \(a=1\) we get the family introduced by \textit{O. Lecacheux} [Advances in number theory, Oxford: Clarendon Press, 293--301 (1993; Zbl 0809.11068)], and \(a=-n+1\) leads to a subfamily of the simplest cubic fields. Taking \(a=n^2+n+1\) we get a new family whose properties are investigated in the present paper. The case \(a=-n^3+1\) will be studied elsewhere.
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1967962
zbMATH Keywords: cyclic cubic field
zbMATH Keywords: simplest cubic field
zbMATH Keywords: fundamental unit
zbMATH Keywords: class number
reviewed by: Veikko Ennola
MaRDI profile type: Publication
cites work: A note on class numbers of the simplest cubic fields
cites work: Class number 3 problem for the simplest cubic fields
cites work: Automorphism polynomials in cyclic cubic extensions
cites work: Q3342674
cites work: Q4235425
cites work: Q4203366
cites work: A Lower Bound for the Class Number of Certain Cubic Number Fields
cites work: Effective Determination of the Decomposition of the Rational Primes in a Cubic Field
cites work: Q4951606
cites work: Q4003835
cites work: The Simplest Cubic Fields
cites work: Class Numbers of the Simplest Cubic Fields
cites work: A family of cubic fields and zeros of 3-adic \(L\)-functions
full work available at URL: s0022-314x(03)00085-4
OpenAlex ID: W2064978295
Recommended article: A Family of RealPn-Tic Fields
Recommended article: Q4824924
Recommended article: Class Numbers of the Simplest Cubic Fields
Recommended article: A note on class numbers of the simplest cubic fields
Recommended article: Units of a family of fields related to the curve \(X_ 1(25)\).
Recommended article: A generalization of simplest number fields and their integral basis
Recommended article: The exponent three class group problem for some real cyclic cubic number fields
Recommended article: On the Construction of Families of Cyclic Polynomials Whose Roots Are Units
Recommended article: Families of cyclic cubic fields
Recommended article: Class number 3 problem for the simplest cubic fields
description: scientific article; zbMATH DE number 1967962