Relationship of irreducible polynomial of prime degree $p$ and the full symmetric group $S_p$

Question

According to Wikipedia, if $f(x)$ is an irreducible polynomial of prime degree $p$ over $\mathbb{Q}$ and it has two nonreal roots, then the Galois group of $f$ is the full symmetric group $S_p$. For instance, a standard example is often $x^3 - 2$ over $\mathbb{Q}$ whose Galois group is $S_3$, i.e. the symmetric group of permutations on $3$ letters. Why then is the general case true?