Kummer–Vandiver conjecture

In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number h_K of the maximal real subfield ${\displaystyle K=\mathbb {Q} (\zeta _{p})^{+}}$ 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),

As of 2011, there is no particularly strong evidence either for or against the conjecture and it is unclear whether it is true or false, though it is likely that counterexamples are very rare.

The class number h of the cyclotomic field ${\displaystyle \mathbb {Q} (\zeta _{p})}$ 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 ${\displaystyle K=\mathbb {Q} (\zeta _{p})^{+}}$ 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.

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