In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. 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

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  • Let p be a prime, and denote the field of p-adic numbers, as usual, by  . Then the Galois group  , where   denotes the algebraic closure of  , is prosolvable. This follows from the fact that, for any finite Galois extension   of  , the Galois group   can be written as semidirect product  , with   cyclic of order   for some  ,   cyclic of order dividing  , and   of  -power order. Therefore,   is solvable.[1]

See also

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References

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  1. ^ Boston, Nigel (2003), The Proof of Fermat's Last Theorem (PDF), Madison, Wisconsin, USA: University of Wisconsin Press