Local-global principle for Galois groups

Is there any sort of local-global principle for Galois groups? For example, given a polynomial $f \in \mathbb{Q}[X]$ and assuming we know $\mathrm{Gal}(f/\mathbb{Q}_p)$ at all primes $p \le \infty$, what can we say about $\mathrm{Gal}(f/\mathbb{Q})$?

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So yeah, you do get a lot of information from local considerations. But you still need to know how to glue the various $D_{\fp}$ together to get the global Galois group.