The General Equation of the nth Degree

Suppose that f ∈ F[x] is an irreducible polynomial of degree n with distinct roots in its splitting field. If f is a “generic” polynomial, or one chosen at random, then we would not expect there to be any algebraic relations among its roots, apart from those given by the elementary symmetric polynomials. So the group of symmetries of the roots should be the full permutation group of degree n. In Exercise 18.10, you saw that the Galois group of an irreducible quartic, chosen at random, always seems to be S ₄. In fact you probably noticed that it is hard to come up with a quartic whose Galois group is not S4. If you had a suitable test you would find the same for n > 4. In this chapter, we want to give a family of examples of degree p, p prime, with Galois group S_p , and prove that indeed the general polynomial of degree n has Galois group S_n . We shall also give a proof of the fundamental theorem of algebra using Galois theory.