ON CLASS NUMBERS INSIDE THE REAL pTH CYCLOTOMIC FIELD

We fix an integer n >= 1, a prime number l with l x 2n and an integers s >= 0. We deal with a prime number p of the form p = 2nl(f)+1. For 0 <= t <= f , let K-t be the real cyclic field of degree l(t) contained in the pth cyclotomic field, and let htbe the class number of K-t. We show that when p(or f) is large enough with respect ton,lands,a prime number r does not divide the ratio h(f )/h(f-(s+1) )when ever r is a primitive root modulo l(2).