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Kummer–Vandiver conjecture

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In mathematics, 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 , the maximal real subfield of the p-th cyclotomic field. Vandiver's conjecture states that p does not divide hK, 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 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] It has been shown by Kurihara that the conjecture is equivalent to a statement in the algebraic K-theory of the integers, namely that

Kn{Z) = 0

whenever n is a multiple of 4.[3] 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.[4]

References

  1. ^ O'Connor, John J.; Robertson, Edmund F., "Henry Schultz Vandiver", MacTutor History of Mathematics Archive, University of St Andrews
  2. ^ *Buhler, Joe; Crandall, Richard; Ernvall, Reijo; Metsänkylä, Tauno; Shokrollahi, M. Amin (2001). "Computational algebra and number theory". Journal of Symbolic Computation. Vol. 31, no. 1–2. pp. 89–96. MR1806208. {{cite news}}: |contribution= ignored (help)
  3. ^ http://mathnet.kaist.ac.kr/mathnet/thesis_file/02-sykim.pdf, p. 4.
  4. ^ http://www.math.harvard.edu/~clarkbar/lichten.pdf, p. 5.

VandiversConjecture at PlanetMath.