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Let $q=p^n$ for some prime $p$. The splitting field of $P\in \mathbb{F}_q[X]$ over $\mathbb{F}_q$ is $\mathbb{F}_{q^m}$ for some integer $m$, and is a Galois extension of $\mathbb{F}_q$. How can I find out about $m$, or the Galois group from the irreducible factors of $P$?

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The degree of $F_{q^m}$ over $F_q$ is $m$, and the Galois group is the cyclic group of $m$ elements. In terms of $P$, $m$ is the least common multiple of the degrees of the irreducible factors of $P$.

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