Générateurs de l'anneau des entiers d'une extension cyclotomique

Abstract

Soit p un nombre premier impair, soit q = p m , où m est un entier positif ; notons ζ q une racine primitive q-ième de l'unité et O q l'anneau des entiers de Q(ζ q ). Dans [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372-384] I. Gaál et L. Robertson montrent que si (h + q , p(p − 1)/2) = 1, où h + q est l'ordre du groupe des classes de Q(ζ q + ζ q ), alors si α ∈ O q engendre O q (autrement dit Z[α] = O q ) soit α est un conjugué d'un translaté par un entier de ζ q soit α + α est un entier impair. Dans notre travail nous montrons que nous pouvons éliminer l'hypothèse sur h + q . Autrement dit nous prouvons que si α ∈ O q engendre O q soit α est un conjugué d'un translaté par un entier de ζ q soit α + α est un entier impair. AbstractLet p be an odd prime and q = p m , where m is a positive integer. Let ζ q be a qth primitive root of 1 and O q be the ring of integers in Q(ζ q ). In [I. Gaál, L. Robertson, Power integral bases in primepower cyclotomic fields, J. Number Theory 120 (2006) 372-384] I. Gaál and L. Robertson show thateither α is equals to a conjugate of an integer translate of ζ q or α + α is an odd integer. In this paper we show that we can remove the hypothesis over h + q . In other words we show that if α ∈ O q is a generator of O q then either α is a conjugate of an integer translate of ζ q or α + α is an odd integer.

Power bases for rings of integers of abelian imaginary fields

“…Actually, the hypothesis on h + q is not necessary. Indeed, in [11], we proved the following result.…”

“…Hence we have generalized the main result of [11] in the case of a cyclotomic field generated by a primitive nth root of unity such that (n, 6) = 1.…”

On power bases for rings of integers of relative Galois extensions

“…When a power basis exists, Gÿory has shown in [11] that up to integer translation there are only finitely many elements that generate a power basis for the ring of integers. As for cyclotomic fields, there is a conjecture by Bremner on the generators for the ring of integers of Q(ζ p ) with p a prime, and a lot of study has been made towards the resolution and the generalization of the conjecture (see 1, 7, [16][17][18][19].…”