# 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 |