Solvable Extensions

Abstract

We now turn to the question of when an arbitrary polynomial equation p(x) = 0 is solvable by radicals. Loosely speaking, this means (for char(F) = 0) that we can reach the roots of p(x) by a finite process of adjoining nth roots of existing elements, that is, by a finite process of passing from a field K to a field K(α), where α is a root of a binomial xⁿ − u, with u ∈ K. We begin with some basic facts about solvable groups.