ABSTRACT
In this chapter, we look at what Galois theory says about quartic equations. First let us recall what we know about cubics. According to Exercise 17.8, the Galois group of an irreducible polynomial acts transitively on its roots. The only transitive subgroups of S 3 are A 3 and S 3 itself. So these are the only possible Galois groups if the cubic f is irreducible. (What if f is reducible?) Furthermore, its Galois group is A 3 if and only if the discriminant Δ is a square in F.
