Kummer–Vandiver conjecture

The Kummer–Vandiver conjecture states that, for every odd prime p, p does not divide the class number of the maximal real subfield of the pth cyclotomic field.

Connect

Cyclotomic field: A field generated over the rationals by a root of unity. The conjecture concerns the cyclotomic field generated by a primitive pth root of unity.
Cyclotomic units: Units in a cyclotomic field generated by roots of unity and their simple expressions. Their index among real units connects cyclotomic arithmetic to class-number factors.
Ernst Kummer: A nineteenth-century German mathematician known for work on number theory and ideal theory. Kummer’s investigations of Fermat’s Last Theorem and cyclotomic fields supplied the conjecture’s background.
Vandiver's conjecture computations: Computational checks of the Kummer–Vandiver conjecture for finite ranges of odd primes. Finite verification provides evidence but cannot establish the assertion for every odd prime.
Maximal real subfield: The largest subfield of a number field contained in the real numbers. This is the specific subfield whose class number appears in the conjecture.
Dirichlet unit theorem: A theorem describing the rank and structure of the units in a number field. It explains the real unit group involved in class-number formulas for cyclotomic fields.
Harry S. Vandiver: An American mathematician who made extensive contributions to number theory and Fermat’s Last Theorem. Vandiver promoted and computationally investigated the real-subfield class-number assertion.
Irregular prime: An odd prime dividing the numerator of at least one Bernoulli number in a specified range. Irregularity concerns the full cyclotomic class number, unlike the conjecture’s real-subfield condition.
Ideal class group: A finite abelian group measuring the failure of unique factorization of ideals in a number field. Its order is the class number whose divisibility the conjecture restricts.
Relative class number: The ratio of a number field’s class number to that of a specified subfield. Cyclotomic class-number formulas separate real and relative contributions.