Applications of Galois Theory

Abstract

In the previous chapters we have developed group theory, field theory, and in particular, Galois theory to a sufficiently high level, allowing us to consider applications to some prominent classical problems. We start in Sect. 6.1 with the problem of solving algebraic equations by radicals , which is the problem that motivated E. Galois to work out his “Galois” theory. In particular, we show for a monic separable polynomial f with coefficients in a field K that the algebraic equation f(x) = 0 is solvable by radicals if and only if the corresponding Galois group is solvable in the group-theoretic sense.