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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 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 , and independently rediscovered around 1920 by Philipp Furtwängler and , 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. ==Background== The class number ''h'' of the cyclotomic field is a product of two integers ''h''1 and ''h''2, called the first and second factors of the class number, where ''h''2 is the class number of the maximal real subfield of the ''p''-th cyclotomic field. The first factor ''h''1 is well understood and can be computed easily in terms of Bernoulli numbers, and is usually rather large. The second factor ''h''2 is not well understood and is hard to compute explicitly, and in the cases when it has been computed it is usually small. Kummer showed that if a prime ''p'' does not divide the class number ''h'', then Fermat's last theorem holds for exponent ''p''. The Kummer–Vandiver conjecture states that ''p'' does not divide the second factor ''h''2. 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kummer–Vandiver conjecture」の詳細全文を読む スポンサード リンク
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