Collinearity equation

2013-11-13T09:43:18Z

137.138.117.212:

In [[mathematics]], more precisely in [[algebra]], a '''prosolvable group''' (less common: '''prosoluble group''') is a [[group (mathematics)|group]] that is [[isomorphic]] to the [[inverse limit]] of an [[inverse system]] of [[solvable group]]s. Equivalently, a group is called '''prosolvable''', if, viewed as a [[topological group]], every [[open neighborhood]] of the identity contains a [[normal subgroup]] whose corresponding [[quotient group]] is a [[solvable group]].

== Examples ==
* Let ''p'' be a [[prime]], and denote the [[Field (mathematics)|field]] of [[p-adic numbers]], as usually, by <math>\mathbf{Q}_p</math>. Then the [[Galois group]] <math>\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)</math>, where <math>\overline{\mathbf{Q}}_p</math> denotes the [[algebraic closure]] of <math>\mathbf{Q}_p</math>, is prosolvable. This follows from the fact that, for any finite [[Galois extension]] <math>L</math> of <math>\mathbf{Q}_p</math>, the [[Galois group]] <math>\text{Gal}(L/\mathbf{Q}_p)</math> can be written as [[semidirect product]] <math>\text{Gal}(L/\mathbf{Q}_p)=(R \rtimes Q) \rtimes P</math>, with <math>P</math> cyclic of order <math>f</math> for some <math>f\in\mathbf{N}</math>, <math>Q</math> cyclic of order dividing <math>p^f-1</math>, and <math>R</math> of <math>p</math>-power order. Therefore, <math>\text{Gal}(L/\mathbf{Q}_p)</math> is [[solvable group|solvable]].<ref>{{citation|last=Boston|first=Nigel|title=The Proof of Fermat's Last Theorem|year=2003|publisher=University of Wisconsin Press|location=Madison, Wisconsin, USA}}</ref>

== See also ==
* [[Solvable group]]
* [[Galois theory]]
* [[p-adic integers]]

==References==
{{reflist}}

[[Category:Mathematical structures]]
[[Category:Algebra]]
[[Category:Number theory]]
[[Category:Topology]]
[[Category:Group theory]]
[[Category:Topological groups]]

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Collinearity equation