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

Ennola, Veikko

J. Théor. Nombres Bordx. 16, No. 3, 569-575 (2004).

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 \(\ell\leq 10000\).

Reviewer: M. E. Keating (London)

MSC:

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

Keywords:

cubic field; fundamental unit; maximal order; non-abelian field; Maple computation

Software:

Maple

References:

[1]	B. N. Delone, D. K. Faddeev, The Theory of Irrationalities of the Third Degree. Trudy Mat. Inst. Steklov, vol. 11 (1940); English transl., Transl. Math. Monographs, vol. 10, Amer. Math. Soc., Providence, R. I., Second printing 1978. · Zbl 0133.30202
[2]	V. Ennola, Cubic number fields with exceptional units. Computational Number Theory (A. Pethö et al., eds.), de Gruyter, Berlin, 1991, pp. 103-128. · Zbl 0732.11054
[3]	H. G. Grundman, Systems of fundamental units in cubic orders. J. Number Theory 50 (1995), 119-127. · Zbl 0828.11061
[4]	M. Mignotte, N. Tzanakis, On a family of cubics. J. Number Theory 39 (1991), 41-49, Corrigendum and addendum, 41 (1992), 128. · Zbl 0763.11011
[5]	E. Thomas, Fundamental units for orders in certain cubic number fields. J. Reine Angew. Math. 310 (1979), 33-55. · Zbl 0427.12005