Vandiver’s conjecture

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).