Power bases for rings of integers of abelian imaginary fields

Abstract

Let L be a number field and let OL be its ring of integers. It is a very difficult problem to decide whether O L has a power basis. Moreover, if a power basis exists, it is hard to find all the generators of O L over Z. In this paper, we show that if α is a generator of the ring of integers of an abelian imaginary field whose conductor is relatively prime to 6, then either α is an integer translate of a root of unity, or α + α is an odd integer. From this result and other remarks it follows that if β is a generator of the ring of integers of an abelian imaginary field with conductor relatively prime to 6 and β is not an integer translate of a root of unity, then ββ is a generator of the ring of integers of the maximal real field contained in Q(β). Finally, we use a result of Gras to prove that if d > 1 is an integer relatively prime to 6, then all but finitely many imaginary extensions of Q of degree 2d have a ring of integers that does not have a power basis.

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].…”