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).

Title	Vandiver’s conjecture
Canonical name	VandiversConjecture
Date of creation	2013-03-22 15:01:11
Last modified on	2013-03-22 15:01:11
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	5
Author	mathcam (2727)
Entry type	Conjecture
Classification	msc 11R29
Related topic	ClassNumbersAndDiscriminantsTopicsOnClassGroups