If $f(x)\in\mathbb{Q}[x]$ of degree $p$ and $\operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$ then $f(x)$ is irreducible. Let $f(x)\in\mathbb{Q}[x]$ , $p$ prime, $\deg f(x)=p$ and $G = \operatorname{Gal}(K/\mathbb{Q})$ has element of order $p$, where $K$ the the splitting field of $f(x)$ over $\mathbb{Q}$.
Show that $f(x)$ is irreducible over $\mathbb{Q}$.

Assume that $\sigma$ is an automorphism of order $p$, also $\sigma$ is a permutation of the roots $a_1, \ldots , a_p$ of $f(x)$. Thus $\sigma$ must permute these roots in a cycle. This means that all the roots are in fact conjugate. Thus $f(x)$ is irreducible.
