Algorithm to determine if a given polynomial over $\mathbb{Q}$ is irreducible over the $p$-adic number field Let $\mathbb{Q}_p$ be the $p$-adic number field.
Let $f(x) \in \mathbb{Q}[x]$ be a polynomial of degree $\ge 1$.

Is there an algorithm to determine whether $f(x)$ is irreducible over $\mathbb{Q}_p$ or not?

The motivation is that there is indeed such an algorithm if we replace $\mathbb{Q}_p$ by $\mathbb{R}$
 A: By googling “p-adic irreducibility”, I found this link to a paper by D. Cantor and D. Gordon. Evidently there are algorithms, but let me point out to you where some the difficulties may lie.
In the real case, it’s easy, ’cause an irreducible polynomial can have degree only $1$ or $2$; and for a quadratic polynomial, you determine irreducibility by looking at the sign of the discriminant.
In the $p$-adic case, there are irreducible polynomials of all degrees, by Eisenstein, which tells you that $x^n-p$ is irred. The general methods of showing reducibility are Hensel’s Lemma in the strong form, which says that if $f(X)\in\mathbb Z_p[X]$ factors into two relatively prime factors as an $\Bbb F_p$-polynomial, then it factors over $\Bbb Z_p$. And the Newton polygon as a general tool, which says that if the (nonvertical part of) the polygon has more than one segment, the polynomial is reducible.
The $p$-adic irreducibility techniques that I’m aware of are, again the Newton polygon, which says that if the polygon’s unique nonvertical segment passes through no integral points in the plane other than its endpoints, then the polynomial is irreducible; and the obvious condition that if the associated $\Bbb F_p$-polynomial is irreducible, then the original $\Bbb Z_p$-polynomial is irreducible. Note that the familiar Eisenstein Criterion for irreducibility is a special case of the Newton-polygon method I mention above.
Finally, note that to apply the second irreducibility criterion I mentioned before, you need to determine whether an $\Bbb F_p$-polynomial is irreducible. This is already a difficult problem.
I leave it at that, and refer you to the Cantor-Gordon paper, and any others that you surely will find when you dig deeper.
