Solvable group

Group with subnormal series where all factors are abelian In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Motivation Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equations. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial f ∈ F [ x ] {\displaystyle f\in F[x]} there is a tower of field extensions F = F 0 ⊆ F 1 ⊆ F 2 ⊆ ⋯ ⊆ F m = K {\displaystyle F=F_{0}\subseteq F_{1}\subseteq F_{2}\subseteq \cdots \subseteq F_{m}=K} such that F i = F i − 1 [ α i ] {\displaystyle F_{i}=F_{i-1}[\alpha _{i}]} where α i m i ∈ F i − 1 {\displaystyle \alpha _{i}^{m_{i}}\in F_{i-1}} , so α i {\displaystyle \alpha _{i}} is a solution to the equation x m i − a {\displaystyle x^{m_{i}}-a} where a ∈ F i − 1 {\displaystyle a\in F_{i-1}} F m {\d