Power integral bases in prime-power cyclotomic fields

Abstract

Let p be an odd prime and q = p m , where m is a positive integer. Let ζ be a primitive qth root of unity, and O q be the ring of integers in the cyclotomic field Q(ζ ). We prove that if O q = Z[α] and gcd(h + q , p(p − 1)/2) = 1, where h + q is the class number of Q(ζ + ζ −1 ), then an integer translate of α lies on the unit circle or the line Re(z) = 1/2 in the complex plane. Both are possible since O q = Z[α] if α = ζ or α = 1/(1 + ζ ). We conjecture that, up to integer translation, these two elements and their Galois conjugates are the only generators for O q , and prove that this is indeed the case when q = 25.

Power bases for rings of integers of abelian imaginary fields

“…Robertson [13] proves the following theorem which shows that the conjecture is true if q is a power of 2. In [3] Gaál and Robertson obtain a result similar to Theorem 1.1 for the classes of generators of Z[ζ q ], where q is a power of an odd prime number p; more precisely they prove the following theorem. is the order of the class group of Q(ζ q + ζ q ), then either α is equivalent to ζ q or α + α is an odd integer.…”

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

“…Alors soit α est équivalent à ζ q , soit α est équivalent à 1/(ζ q + 1). Robertson [8] démontre le théorème suivant qui preuve une telle conjecture dans le cas où q est une puissance de 2 : Ensuite, Robertson et Gaál [3] obtiennent un résultat similaire au théorème 1 pour les classes des générateurs de Z[ζ q ], où q est la puissance d'un nombre premier p, plus précisément :…”