Vandiver's conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For each prime $p$, set $\zeta_p=\exp(2\pi i/p)$ and $K_p^+=\mathbb Q(\zeta_p+\zeta_p^{-1})=\mathbb Q(\zeta_p)\cap\mathbb R$. This is the maximal real subfield of the $p$th cyclotomic field. Its ideal class group is the group of nonzero fractional ideals of its ring of integers, modulo principal fractional ideals; its order $h_p^+$ is the class number.

Statement recorded in the supplied source.

Is $p\nmid h_p^+$ for every prime $p$? Equivalently, is the $p$-primary subgroup of the ideal class group of $K_p^+$ trivial for every prime $p$? [S2]

Short English statement

Does a prime never divide the class number of the maximal real subfield of its cyclotomic field?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8900006 (AMR-088-0006). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Christian Wuthrich, Open problems in number theory, lecture slides (December 2011), slide headed "Vandiver's conjecture, PDF page 31. https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf.