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 OL has a power basis. Moreover, if a power basis exists, it is hard to find all the generators of OL over Z. In this paper, we show that if a is a generator of the ring of integers of an abelian imaginary field whose conductor is relatively prime to 6, then either a is an integer translate of a root of unity, or a + a is an odd integer. From this result and other remarks it follows that if â is a generator of the ring of integers of an abelian imaginary field with conductor relatively prime to 6 and â is not an integer translate of a root of unity, then ââ is a generator of the ring of integers of the maximal real field contained in Q(â). 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.