Vandiver's conjectureVandiversConjecture2005-02-08 20:08:162008-05-27 16:22:20Conjecture% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}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).