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Local Area Info: Kummer–Vandiver conjecture

In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number hK of the maximal real subfield K = Q ( ? p ) + {displaystyle K=mathbb {Q} (zeta _{p})^{+}} of the p-th cyclotomic field. The conjecture was first made by Ernst Kummer in 1849 December 28 and 1853 April 24 in letters to Leopold Kronecker, reprinted in (Kummer 1975, pages 84, 93, 123–124), and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver (1946,?p. 576),

The class number h of the cyclotomic field Q ( ? p ) {displaystyle mathbb {Q} (zeta _{p})} is a product of two integers h1 and h2, called the first and second factors of the class number, where h2 is the class number of the maximal real subfield K = Q ( ? p ) + {displaystyle K=mathbb {Q} (zeta _{p})^{+}} of the p-th cyclotomic field. The first factor h1 is well understood and can be computed easily in terms of Bernoulli numbers, and is usually rather large. The second factor h2 is not well understood and is hard to compute explicitly, and in the cases when it has been computed it is usually small.

The Kummer–Vandiver conjecture states that p does not divide the second factor h2. Kummer showed that if p divides the second factor, then it also divides the first factor. In particular the Kummer–Vandiver conjecture holds for regular primes (those for which p does not divide the first factor).