Fundamental units in a family of cubic fields. (English) Zbl 1079.11056

E. Thomas has shown that, for a suitable choice of a root \(\varepsilon\) of the polynomial \[ x^3+ (\ell-1) x^2- \ell x- 1,\quad \ell\in\mathbb{Z},\quad \ell\geq 3, \] \(\varepsilon\), \(\varepsilon- 1\) is a fundamental pair of units for the order \(\mathbb{Z}[\varepsilon]\). This note gives a criterion for this pair to be also fundamental for the maximal order \({\mathcal O}\) of the non-abelian field \(\mathbb{Q}(\varepsilon): [{\mathcal O}: \mathbb{Z}[\varepsilon]]\leq \ell/3\). A Maple computation verifies that this inequality holds for all \(\ell\leq 10000\).


11R27 Units and factorization
11R16 Cubic and quartic extensions
11Y40 Algebraic number theory computations


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