Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathrm{Gal}(f/\mathbb{Q})$?

share|improve this question

1 Answer 1

up vote 10 down vote accepted

$\newcommand\cO{\mathcal{O}}$ $\newcommand\fp{\mathfrak{p}}$ $\newcommand\Q{\mathbb{Q}}$ 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.