NOTE ON THE CLASS NUMBER OF THE pTH CYCLOTOMIC FIELD

Let p be a prime number of the form p = 2l(f) + 1 with an odd prime number l, and h(p)(-) the relative class number of the pth cyclotomic field K = Q(zeta(p)). When f = 1, it is conjectured that h(p)(-) is odd, and there are several results related to this conjecture. In this paper, we deal with the case f >= 2. For 0 <= t <= f, let h(p,t) denote the relative class number of the imaginary subfield K-t of K of degree 2l(t) over Q. We show that the ratio h(p,f)(-)/h(p,f-1)(-) is not divisible by a prime number r if r is a primitive root modulo l(2). Further, when r <= 47, we give some computational results on the ratio h(p,t)(-)/h(p,t-1)(-) for 1 <= t <= f. In the range of our computation, we find that the ratio is divisible by r only in some exceptional cases.