Let $P\in\mathbb{Q}[X]$ be some polynomial. Let $p$ be some prime number. We can consider $P\in\mathbb{Q}_p[X]$. What informations can we get from the Galois group of $P\in\mathbb{Q}_p[X]$ to the Galois group of $P\in\mathbb{Q}[X]$? I imagine that only one $p$ is not sufficient for that so I guess that the correct question should be: what informations can we get from the set of all Galois group of $P\in\mathbb{Q}_p[X]$, for all $p$, to the Galois group of $P\in\mathbb{Q}[X]$?
For example the fact that $P$ is irreducible in one $\mathbb{Q}_p$ imply that it is irreducible in $\mathbb{Q}$ so that the transitivity of the action of $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ on the roots of $P\in\mathbb{P}_p[X]$ imply the transitivity of the action of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the roots of $P\in\mathbb{Q}[X]$.
