As I'm new into p-adic Field Theory, I learned Hensel's Lemma by which I can show if a polynomial has roots over the p-adic numbers (i.e. in $\mathbb{Q}_2$ or $\mathbb{Q}_3$) . For polynomials of degree 2 or 3 over those fields, with Hensels Lemma I can show they are irreducible over those fields if there aren't any roots in $\mathbb{Q}_2$ or $\mathbb{Q}_3$.
But how can I show a polynomial of degree 4 is irreducible over such a field? Since for degree 4, its not sufficient to show there are no roots in $\mathbb{Q}_2$ or $\mathbb{Q}_3$. Is there an easy way to show it? Or is there a polynomial of degree 4 which obviously is irreducible in general over such a p-adic Field?