Concept

Solvable group

Summary
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 equation. 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 \in F[x] there is a tower of field extensionsF = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=Ksuch that

F_i = F_{i-1}[\alpha_i] where \alpha_i^{m_i} \in F_{i-1}, so \alpha_i is a solution to the equation x^{m_i} - a where a \in F_{i-1}

F_m con

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