Detecting fast solvability of equations via small powerful Galois groups
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- by S. K. Chebolu, J. Mináč and C. Quadrelli PDF
- Trans. Amer. Math. Soc. 367 (2015), 8439-8464 Request permission
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.References
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Additional Information
- S. K. Chebolu
- Affiliation: Department of Mathematics, Illinois State University, Campus box 4520, Normal, Illinois 61761
- Email: schebol@ilstu.edu
- J. Mináč
- Affiliation: Department of Mathematics, University of Western Ontario, Middlesex College, London, Ontario N6A5B7, Canada
- Email: minac@uwo.ca
- C. Quadrelli
- Affiliation: Dipartimento di Matematica, Università di Milano-Bicocca, Ed. U5, Via R.Cozzi 53, 20125 Milano, Italy
- Email: c.quadrelli1@campus.unimib.it
- 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 - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 8439-8464
- MSC (2010): Primary 12F10, 12G10, 20E18
- DOI: https://doi.org/10.1090/S0002-9947-2015-06304-1
- MathSciNet review: 3403061
Dedicated: Dedicated to Professors Tsit-Yuen Lam and Helmut Koch with admiration and respect