 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 formula_1 there is a tower of field extensionsformula_2such that


For example, the smallest Galois field extension of formula_10 containing the elementformula_11gives a solvable group.