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# 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).

Related:

ClassNumbersAndDiscriminantsTopicsOnClassGroups

Type of Math Object:

Conjecture

Major Section:

Reference

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## Mathematics Subject Classification

11R29*no label found*

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## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

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new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag