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 GF (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 GF (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.

Chebolu, S., Minac, J., Quadrelli, C. (2015). Detecting fast solvability of equations via small powerful galois groups. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 367(12), 8439-8464 [10.1090/S0002-9947-2015-06304-1].

Detecting fast solvability of equations via small powerful galois groups

QUADRELLI, CLAUDIO
2015

Abstract

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 GF (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 GF (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.
Articolo in rivista - Articolo scientifico
Rigid fields, Galois modules, absolute Galois groups, Bloch-Kato groups, powerful pro-p groups
English
2015
367
12
8439
8464
reserved
Chebolu, S., Minac, J., Quadrelli, C. (2015). Detecting fast solvability of equations via small powerful galois groups. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 367(12), 8439-8464 [10.1090/S0002-9947-2015-06304-1].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/47580
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