Following Serre´s question, the Sylow subgroups of the absolute Galois group of a p-adic field were studied and completely understood by Labute. However, the structure of the p-Sylow subgroups of the absolute Galois group of the field of rational numbers is much more subtle and mysterious. We make the first steps towards its determination via a surprisingly simple decomposition. Joint work with Lior Bary Soroker and Moshe Jarden.