Power integral bases in prime-power cyclotomic fields Original Research Article

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 image be the ring of integers in the cyclotomic field image. We prove that if image and image, where image is the class number of image, 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 image 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 image, and prove that this is indeed the case when q=25.