A condition for divisibility of the class number of real pth cyclotomic field by an odd prime distinct from p

Abstract

Let p > 3 be an odd prime. Let ~" = ~'p = cos(2zr/p) + i s in(2rr/p) be a primitive pth root of unity. Let Q(~-p)+ denote the maximal real subfield of the pth cyclotomic field Q(~'p), i.e., Q(~'p)+ = Q(~'p § ~-pl). Let h + be the class number of Q(ffp)+. The parity criterion of h + is studied in several papers [1, 2, 14, 21]. However, very little is known about h +. Let ~ be an odd prime distinct from p. Then it is interesting to study when h + is divisible by ~. KUMMER [14] investigated this problem and gave a sufficient condition for h + to be indivisible by (Satz VI). When p = 2q § 1 and q is also a prime, JAKUBEC [9] shows that if is a primitive root modulo q, then ~ does not divide h + and in [12] that if q = 3 (mod 4) and the order o f s modulo q is (q 1)/2, then h + is indivisible by/~ 0 for any prime p, 3 < p < 700 and for any prime e # p, 3 < ~ < 31. That is, for each odd prime ~ < 31, we shall tabulate all the primes