  Surprisingly, the class numbers of cyclotomic fields have only been
determined for fields of small conductor, e.g. for prime conductors up to 67,
due to the problem of finding the "plus part," i.e. the class number of the
maximal real subfield. Our results have improved the situation. We prove that
the plus part of the class number is 1 for prime conductors between 71 and 151.
Also, under the assumption of the generalized Riemann hypothesis, we determine
the class number for prime conductors between 167 and 241. This technique
generalizes to any totally real field of moderately large discriminant,
allowing us to confront a large class of number fields not previously
treatable.
