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Let $f \in \mathbb{Q}[X]$ be irreducible and let $L$ be its splitting field. Can something be said about the Galois group of $L$ over $\mathbb{Q}$ without computing the roots of $f$ in $\mathbb{C}$?

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    Would any of you care to elaborate in an answer? It's strange if $G$ can be determined completely, since whenever I look fr examples I see automorphisms written explicitly using the roots. Also, isn't Chebotarev density theorem a thing of probability? How does that help?2012-09-14

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