Kummer–Vandiver conjecture

Let $p$ be an odd prime, let $\zeta_p$ be a primitive $p$-th root of unity, and let

$$K = \mathbb{Q}(\zeta_p)^{+} = \mathbb{Q}(\zeta_p + \zeta_p^{-1})$$

be the maximal real subfield of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$. Let $h_K$ denote the class number of $K$, i.e. the order of the ideal class group of the ring of integers of $K$; equivalently, $h_K = h_2$ is the second factor in the decomposition $h = h_1 h_2$ of the class number $h$ of $\mathbb{Q}(\zeta_p)$, where $h_2 = h_K$ and $h_1 = h/h_K$.

Then

$$p \nmid h_K$$

for every odd prime $p$.