Detecting fast solvability of equations via small powerful galois groups

Fix an odd prime p, and let F be a field containing a primitive pth root of unity. It is known that a p-rigid field F is characterized by the property that the Galois group G_F (p) of the maximal p-extension F(p)/F is a solvable group. We give a new characterization of p-rigidity which says that a field F is p-rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic p-adic groups and to some Galois modules. When F is p-rigid, we also show that it is possible to solve for the roots of any irreducible polynomials in F[X] whose splitting field over F has a p-power degree via non-nested radicals. We provide new direct proofs for hereditary p-rigidity, together with some characterizations for G_F (p) - including a complete description for such a group and for the action of it on F(p) - in the case F is p-rigid.