Kummer–Vandiver conjecture

In mathematics, Vandiver's conjecture concerns a property of algebraic number fields. Although attributed to American mathematician Harry Vandiver, 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. It has been shown by Kurihara that the conjecture is equivalent to a statement in the algebraic K-theory of the integers, namely that

K_n{Z) = 0

whenever n is a multiple of 4. In fact from Vandiver's conjecture follows a full conjectural calculation of the K-groups for all values of n; so that it amounts to the same to assume the "expected" calculation of those K-groups. This was a conditional result, and is now a consequence of the norm residue isomorphism theorem.

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

VandiversConjecture at PlanetMath.