Class numbers of cyclotomic fields

AbstractIn the first part of the paper we show how to construct real cyclotomic fields with large class numbers. If the GRH holds then the class number hp+ of the pth real cyclotomic field satisfies hp+ > p for the prime p = 11290018777. If we allow n to be composite we have, unconditionally, that hn+ > n32 − ε for infinitely many n. In the second part of the paper we show that if l ≢= 2 mod 4 and n is the product of 4 distinct primes congruent to 1 mod l, then l2 (l, if l is odd) divides the class number hn+ of the nth cyclotomic field. If the primes are congruent to 1 mod 4l then 2l divides hn+