The Sylow subgroups of the absolute Galois group [formula omitted

Abstract

We describe the ℓ-Sylow subgroups of Gal(Q) for an odd prime ℓ, by observing and studying their decomposition as F⋊Zℓ, where F is a free pro-ℓ group, and Zℓ are the ℓ-adic integers. We determine the finite Zℓ-quotients of F and more generally show that every split embedding problem of Zℓ-groups for F is solvable. Moreover, we analyze the Zℓ-action on generators of F.