On power bases in cyclotomic number fields

The current paper considers the question of power bases in the cyclotomic number field Q(ζ), ζ^p = 1, p an odd prime. The ring of integers is Z[ζ], and there do exist further "non-obvious" generators for this order; specifically we shall see that Z[α] = Z[ζ] for α = ζ + ζ² + ⋯ + ζ ^{(p-1) 2}. We conjecture that, up to conjugacy, there can be no further such integral generators, and prove that this is indeed the case in Q(ζ₇).