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Detecting fast solvability of equations via small powerful Galois groups

Authors: S. K. Chebolu, J. Mináč and C. Quadrelli
Journal: Trans. Amer. Math. Soc. 367 (2015), 8439-8464
MSC (2010): Primary 12F10, 12G10, 20E18
Published electronically: April 1, 2015
MathSciNet review: 3403061
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Abstract: Fix an odd prime $ p$, and let $ F$ be a field containing a primitive $ p$th 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.

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Additional Information

S. K. Chebolu
Affiliation: Department of Mathematics, Illinois State University, Campus box 4520, Normal, Illinois 61761

J. Mináč
Affiliation: Department of Mathematics, University of Western Ontario, Middlesex College, London, Ontario N6A5B7, Canada

C. Quadrelli
Affiliation: Dipartimento di Matematica, Università di Milano-Bicocca, Ed. U5, Via R.Cozzi 53, 20125 Milano, Italy

Keywords: Rigid fields, Galois modules, absolute Galois groups, Bloch-Kato groups, powerful pro-$p$ groups
Received by editor(s): June 13, 2012
Received by editor(s) in revised form: September 17, 2013
Published electronically: April 1, 2015
Additional Notes: The first author was partially supported by NSA grant H98230-13-1-0238
The second author was partially supported by NSERC grant RO37OA1OO6
The third author was partially supported by an INDAM-GNSAGA travel grant
Dedicated: Dedicated to Professors Tsit-Yuen Lam and Helmut Koch with admiration and respect
Article copyright: © Copyright 2015 American Mathematical Society