Djordjo Milovic, Université Paris Sud et Leiden (Pays-Bas)

The infinitude of Q(p^1/2) with class number divisible by 16

The density of primes p such that the class number h of Q(p^1/2) is divisible by 2^k is conjectured to be 2^-k for all positive integers k. The conjecture is true for 1 ≤ k ≤ 3 but still open for k ≥ 4. For primes p of the form a² + c⁴ with c even, we find the 8-Hilbert class field of Q(p^1/2) in terms of a and c. We then use a theorem of Iwaniec and Friedlander to show that there are infinitely many primes p for which h is divisible by 16, and also infinitely many primes p for which h is divisible by 8 but not by 16.