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Surprisingly, the class numbers of cyclotomic fields have only beendetermined 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 themaximal real subfield. Our results have improved the situation. We prove thatthe 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 determinethe class number for prime conductors between 167 and 241. This techniquegeneralizes to any totally real field of moderately large discriminant,allowing us to confront a large class of number fields not previouslytreatable.

Real cyclotomic fields of prime conductor and their class numbers