Detecting fast solvability of equations via small powerful Galois groups

Abstract

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