Abstract

We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two well-known and widely believed conjectures of analytic number theory. 1. Introduction In 1850 Kummer [13] published a review of the main results that he and others had discovered about cyclotomic fields. In this elegant report he claimed that he had found an explicit "law for the asymptotic growth" of h 1 (p), the so-called first factor of the class number of the cyclotomic field, and would provide a proof elsewhere. This proof never appeared and we believe that Kummer's claim is incorrect. More precisely, let p denote any odd prime, let h(p) be the class number of the cyclotomic field Q(i p ) (where i p is a primitive pth root of unity) and h 2 (p) be the class number of the real subfield Q(i p +i \Gamma1 p ). Kummer proved that the ratio h 1 (p) = h(p)=h 2 (p) is an integer which he called the first factor of the ...

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