Galois Groups of Cubics and Quartics

Let K be a field and f(X) be a separable polynomial in K[X]. The Galois group of f(X) over K permutes the roots of f(X) in a splitting field, and labeling the roots as r1, . . . , rn provides an embedding of the Galois group into Sn. We recall without proof two theorems about this embedding. Theorem 1.1. Let f(X) ∈ K[X] be a separable polynomial of degree n. (a) If f(X) is irreducible in K[X] then its Galois group over K has order divisible by n. (b) The polynomial f(X) is irreducible in K[X] if and only if its Galois group over K is a transitive subgroup of Sn. Definition 1.2. If f(X) ∈ K[X] factors in a splitting field as f(X) = c(X − r1) · · · (X − rn), the discriminant of f(X) is defined to be