A recursive description of pro-p Galois groups

n this note we study a modified version of the “Elementary Type Conjecture” for pro-p Galois groups. To be precise, let Gp(F) be the Galois group of the maximal Galois p-extension of a field F containing a primitive pth root of unity. Under some natural assumptions concerning valuation rings of F (and also orderings of F when p=2), we prove that Gp(F) can be obtained from suitable closed subgroups using a finite number of free pro-p products and semidirect products.