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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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$\newcommand\cO{\mathcal{O}}$ $\newcommand\fp{\mathfrak{p}}$ $\newcommand\Q{\mathbb{Q}}$ Yes, there is. Let $K$ be your splitting field over $\Q$, denote its ring of integers by $\cO$, and the Galois group by $G$. For any rational prime $p$, fix a prime $\fp$ of $\cO$ over $p$. Denote by $D_{\fp}$ the subgroup of $G$ that fixes $\fp$. This is called the decomposition group of $\fp$. Then $D_{\fp}$ is identified with the Galois group of the completion of $K$ at $\fp$ over $\Q_p$, which is precisely the Galois group of $f$ over $\Q_p$ (the primes above $p$ are all $G$-conjugate to each other, so the groups $D_{\fp}$ are all conjugate, and thus well-defined up to isomorphism, and in fact up to inner automorphism of $G$). So if you know $D_{\fp}$ for all $\fp$, then you know lots of subgroups of $G$. Moreover, the groups $D_{\fp}$ together generate all of $G$, this is a consequence of Chebotarev's density theorem.

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.

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