A family of cyclic cubic polynomials whose roots are systems of fundamental units. (Q1403936)

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.