ResumeEnnola. 65KB Jun 04 2011 12:07:06 AM

Journal de Th´

eorie des Nombres

de Bordeaux

16

_{(2004), 569–575}

Fundamental units in a family of cubic fields

par

Veikko ENNOLA

R´esum´e. _{SoitO l’ordre maximal du corps cubique engendr´e par} une racineε de l’equationx3

+ (ℓ−1)x2₋

ℓx−1 = 0, o`u ℓ∈<sub>Z</sub>, ℓ≥3. Nous prouvons queε, ε−1 forment un syst`eme fondamental d’unit´es dansO, si [O:Z[ε]]≤ℓ/3.

Abstract. _Let O be the maximal order of the cubic field gen-erated by a zero ε of x3

+ (ℓ−1)x2

−ℓx−1 forℓ ∈ Z, ℓ ≥ 3. We prove that ε, ε−1 is a fundamental pair of units for O, if [O:Z[ε]]≤ℓ/3.

VeikkoEnnola

Department of Mathematics University of Turku FIN-20014, Finland

E-mail_:ennola@utu.fi