Vandiver’s Conjecture

Vandiver's conjecture asserts that for every prime $p$, the class number $h_p^+$ of the maximal real subfield $\mathbb Q(\zeta_p+\zeta_p^{-1})$ of the $p$-th cyclotomic field is not divisible by $p$. The statement is traditionally attributed to Vandiver around 1929, close in time to his paper on singular integers in properly irregular cyclotomic fields [Vandiver1929], though the exact date at which the modern class-number formulation crystallised is not securely documented.

The conjecture sits at the heart of cyclotomic theory, and standard references such as Washington's book develop its many consequences for the structure of class groups of cyclotomic fields [Washington1997Cyclotomic]. Numerically it stands on very firm ground: the computations of Buhler and Harvey verified it for all primes up to $163$ million in the course of tabulating irregular primes [BuhlerHarvey2011Irregular], and heuristic models predict that counterexamples, if any exist, should be extraordinarily rare. Nevertheless no theoretical argument covers all primes at once, and a single prime with $p\mid h_p^+$ would refute the conjecture. It remains open, and a proof would demand presently unavailable structural control of the real cyclotomic class group.