ABSTRACT
30.1 In this chapter, our work finally pays off. Via the abstraction of the field theory we have developed, viewing the Galois group as a group of permutations of the roots of a given polynomial that we want to solve, we actually arrive at effective methods of finding these solutions in several cases. The fact whether the Galois group acts in a transitive way on the set of roots of a polynomial is related to the irreducibility of that polynomial (Theorem 30.5). We include Galois’ Theorem on Natural Irrationalities. One of the really useful tools in solving equation is the discriminant {30.17); in applications it is often used that the discriminant is a square if and only if the Galois group of the polynomial consists of even permutations (Proposition 30.18, note the slight restriction in this proposition, e.g. char K ≠ 2). We provide several examples, see also the Chapter: Exercises. We finally also get rid of the haunting problem of having to decide whether some polynomial is solvable by radicals (look at the Historical Comments to get an idea of how this problem has kept many scientists busy over many centuries). Indeed the combination of Theorem 30.33 and Theorem 30.38 establishes that a polynomial is solvable by radicals if and only if its Galois group is a solvable group. This then provides the method to prove that the general equation of degree n is not solvable if n ≥ 5 (Ruffini, Abel Theorem 30.41). Finally, let us stress this fact again, it is the fruit of a suitable abstraction of an originally very concrete problem concerning the solution of equations to a problem about elements of fields and a group acting on them, that allows to translate the original problem into a verifiable question about the symmetric group Sn , in particular S5 . We hope 550this convinces the student of the applicability of abstract techniques and provides a glimpse of insight in the beauty of abstract mathematics.
