The Galois group of an irreducible polynomial is a transitive subgroup

Let \(k\) be a field, \(f \in k[x]\) irreducible, \(\SF(f)\) a splitting field of \(f\) over \(k\), and \(\alpha_1, \ldots, \alpha_m\) the distinct roots of \(f\) in \(\SF(f)\). Then \(\Gal(\SF(f)/k)\) acts faithfully and transitively on \(\theset{\alpha_1, \ldots, \alpha_m}\), so it is isomorphic to a transitive subgroup of \(S_m\). If \(f\) is separable of degree \(n\), then \(m = n\).