Q:

Irreducible over the rationals

Show that $x^2-4x+2$ is irreducible over the rationals. 
I know I must use Eisenstein's criterion but I don't know how to apply it here.  

A:

Eisenstein's criterion is directly applicable with $p = 2$.

For an alternative proof, it suffices to show that (since the degree of the polynomial is $2$) that there are no rational roots.  This can be done with the rational roots theorem, or by noting that the polynomial splits over $\mathbb{R}$, and its roots are
$$\frac{4 \pm \sqrt{8}}{2} = 2 \pm \sqrt 2$$
Neither is rational.