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Systems of fundamental units in cubic orders. (English) Zbl 0828.11061

H. J. Stender found infinitely many irreducible polynomials \(\varphi (X) = X^3 + eX^2 + fX + g\) of \(\mathbb{Z} [X]\) having three real roots and involving two (resp. three) parameters such that \(\{\lambda + r,\;\lambda + s\}\) (resp. \(\{(\lambda + kr)/ \lambda,\;(\lambda + ks)/ \lambda\})\) is a fundamental system of units of \(\mathbb{Z} [\lambda]\) [see J. Reine Angew. Math. 257, 151-178 (1972; Zbl 0247.12008)]. E. Thomas exhibited a fundamental system of the form \(\{r \lambda + 1,\;\lambda + d\}\) where \(r \geq 2\) [see J. Reine Angew. Math. 310, 33-55 (1979; Zbl 0427.12005)].
In the paper under review, the author exhibits other irreducible polynomials \(\varphi (X)\) with roots \(\lambda = \lambda^{(0)} > \lambda^{(1)} > \lambda^{(2)}\) for which he can exhibit a fundamental system \(S\) of units of \(\mathbb{Z} [\lambda]\) containing two units, each of which being a polynomial in \(\lambda\) of degree 2; in fact, the description of the units is made by considering the continued fraction expansions of \(-\lambda^{(1)}\) and \(- \lambda^{(2)}\). A result of Berwick is used in the proof. As an application, the author exhibits the polynomials \(\varphi (X)\) and the integers \(r \neq 1\) for which \(\{r \lambda + 1,2 \lambda + 3\}\) is a fundamental system of units of \(\mathbb{Z} [\lambda]\). This paper will certainly interest number theorists involved with parametrized units.

MSC:

11R27 Units and factorization
11R16 Cubic and quartic extensions
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