## 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

### Citations:

Zbl 0247.12008; Zbl 0427.12005
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