Vandiver's conjecture

In mathematics, Vandiver's conjecture concerns a property of algebraic number fields. Although attributed to American mathematician Harry Vandiver, [MacTutor Biography|id=Vandiver|title=Henry Schultz 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. [*Citation | last=Buhler | first=Joe | last2=Crandall | first2=Richard | last3=Ernvall | first3=Reijo | last4=Metsänkylä | first4=Tauno | last5=Shokrollahi | first5=M. Amin | contribution=Irregular primes and cyclotomic invariants to 12 million | title=Computational algebra and number theory | periodical=Journal of Symbolic Computation | id=MathSciNet | id = 1806208 | year=2001 | volume=31 | issue=1-2 | pages=89-96]

References

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* 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