Detecting Fast solvability of equations via small powerful Galois groups

2013/10/28 by Sunil K. Chebolu, Chebolu, Sunil K., Jan Minac +3
Mathematics · #12F10 #12G10 #20E18 #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #math.AC #math.AT #math.NT #msc:12F10 #msc:12G10 #msc:20E18

paper · pdf · doi:10.48550/arxiv.1310.7623

25 pages, to appear in "Transactions of the AMS"

arxiv created 2013/10/30 · arxiv updated 2013/10/31

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 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.

Related

• Solvable primitive extensions
• The Kernel Unipotent Conjecture and the vanishing of Massey products for odd rigid fields
• Relations in the maximal pro-p quotients of absolute Galois groups
• The finitude of tamely ramified pro-p extensions of number fields with cyclic p-class groups
• Finding Galois splitting models to compute local invariants