Prosolvable group

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 ${\displaystyle \mathbf {Q} _{p}}$. Then the Galois group ${\displaystyle {\text{Gal}}({\overline {\mathbf {Q} }}_{p}/\mathbf {Q} _{p})}$, where ${\displaystyle {\overline {\mathbf {Q} }}_{p}}$ denotes the algebraic closure of ${\displaystyle \mathbf {Q} _{p}}$, is prosolvable. This follows from the fact that, for any finite Galois extension ${\displaystyle L}$ of ${\displaystyle \mathbf {Q} _{p}}$, the Galois group ${\displaystyle {\text{Gal}}(L/\mathbf {Q} _{p})}$ can be written as semidirect product ${\displaystyle {\text{Gal}}(L/\mathbf {Q} _{p})=(R\rtimes Q)\rtimes P}$, with ${\displaystyle P}$ cyclic of order ${\displaystyle f}$ for some ${\displaystyle f\in \mathbf {N} }$, ${\displaystyle Q}$ cyclic of order dividing ${\displaystyle p^{f}-1}$, and ${\displaystyle R}$ of ${\displaystyle p}$-power order. Therefore, ${\displaystyle {\text{Gal}}(L/\mathbf {Q} _{p})}$ is solvable.^[1]

• Galois theory