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 $h_K$ , the class number of $K$ .

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 $p$ -rank of the ideal class group of $\mathbb{Q}(\zeta_p)$ equals the number of Bernoulli numbers divisible by $p$ (a remarkable strengthening of Herbrand's theorem).