TITLE: Class number of real maximal subfield of cyclotomic fields
QUESTION [9 upvotes]: Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$?
Are there infinitely many? Finitely many? Something else?
(As usual, $\zeta_p$ denotes a primitive $p$-th root of unity.)

REPLY [5 votes]: I thought it should be mentioned that van der Linden's result (which depends on Odlyzko's discriminant bounds) has recently been improved upon by a more careful accounting of the contribution of prime ideals to the explicit formula of the Dedekind zeta function of the Hilbert class field of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$.
The value of $h_p^+$ is now known up to 151 (and up to 241 under GRH).  Please see http://arxiv.org/abs/1407.2373.