Vandiver's conjecture

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Vandiver's conjecture concerns a property of algebraic number fields. Although attributed to American mathematician Harry Vandiver [1], the conjecture was first made in a letter from Ernst Kummer to Leopold Kronecker.

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 entry on regular 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 the Herbrand-Ribet theorem).

Vandiver's conjecture has been verified for p < 12 million. [2]

[edit] References

Lawrence C. Washington (1996). Introduction to Cyclotomic Fields. Springer. ISBN 0-387-94762-0.
E. Ghate, Vandiver's Conjecture via K-theory, 1999 - a survey of work by Soulé and Kurihara - (DVI file) http://www.math.tifr.res.in/~eghate/vandiver.dvi

This article incorporates material from VandiversConjecture on PlanetMath, which is licensed under the GFDL.