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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

Groups audience:

## Mathematics Subject Classification

11R29*no label found*

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