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.
- 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.