A family of cyclic cubic polynomials whose roots are systems of fundamental units. (English) Zbl 1034.11060

Kishi, Yasuhiro

J. Number Theory 102, No. 1, 90-106 (2003); erratum ibid. 103, No. 1, 132-133 (2003). (2003).

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 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.

Reviewer: Veikko Ennola (Turku)

MSC:

11R16	Cubic and quartic extensions
11R27	Units and factorization
11R29	Class numbers, class groups, discriminants

Keywords:

cyclic cubic field; simplest cubic field; fundamental unit; class number

Citations:

Zbl 0809.11068

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