Power integral bases in prime-power cyclotomic fields

Let p be an odd prime and q=pm, where m is a positive integer. Let ζ be a primitive qth root of unity, and Oq be the ring of integers in the cyclotomic field Q(ζ). We prove that if Oq=Z[α] and , where 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 Oq=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 Oq, and prove that this is indeed the case when q=25.