Abstract :
Letpbe an odd prime and imagepbe the ring of integers in the cyclotomic fieldQ(ζ), whereζis a primitivepth root of unity. Then imagep=Z[α] ifα=ζ, 1/(1+ζ), or one of the conjugates of these two elements. In 1988, Bremner [3] conjectured that up to integer translation there are no further generators for imagepand proved that this is indeed the case whenp=7. We establish a criterion for verifying Bremnerʹs conjecture for a given regular primepand use it to prove the conjecture forpless-than-or-equals, slant23,p≠17. A key step in the proof of the criterion is a determinant formula for the relative class numberh−ofQ(ζ).
