Vandiver’s conjecture

Let K=(ζp)+, the maximal real subfield of the p-th cyclotomic fieldMathworldPlanetmath. Vandiver’s conjecture states that p does not divide hK, the class numberMathworldPlanetmathPlanetmath of K.

For comparison, see the entries on regular primesMathworldPlanetmath and irregular primes.

A proof of Vandiver’s conjecture would be a landmark in algebraic number theoryMathworldPlanetmath, 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 (ζp) equals the number of Bernoulli numbersDlmfDlmfMathworldPlanetmathPlanetmath 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