Let , the maximal real subfield of the -th cyclotomic field. Vandiver's conjecture states that does not divide , the class number of .

For comparison, see the entries on regular primes 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 -rank of the ideal class group of equals the number of Bernoulli numbers divisible by (a remarkable strengthening of Herbrand's theorem).