User:Noosfractal/Vandiver's conjecture

Vandiver's conjecture concerns a property of algebraic number fields. Although attributed to American mathematician Harry Vandiver, the conjecture was first made in a letter from Ernst Kummer to Leopold Kronecker.


 * Let $$K=\mathbb{Q}(\zeta_p)^+$$, the maximal real subfield of the p-th cyclotomic field. Vandiver's conjecture states that p does not divide hK, the class number of K.

For comparison, see the entry on regular and irregular primes.

A proof of Vandiver's conjecture would be a landmark in algebraic number theory, as many theorems hinge on the assumption that this conjecture is true. For example, it is known that if Vandiver's conjecture holds, that the p-rank of the ideal class group of $$\mathbb{Q}(\zeta_p)$$ equals the number of Bernoulli numbers divisible by p (a remarkable strengthening of the Herbrand-Ribet theorem).

Vandiver's conjecture has been verified for p < 12 million.