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Let $f$ be a polynomial over a finite field $F$ which decomposes into a product of irreducible factors $f=p_1...p_k$ of degree $n_1,...n_k$. How can I prove that the degree of splitting field of $f$ over $F$ is least common multiple of $n_1,...,n_k$?

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  • $\begingroup$ Could you restate the question? $\endgroup$ – 9301293 Sep 6 '15 at 23:16
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    $\begingroup$ What is wrong with current statement? $\endgroup$ – Ultra Sep 7 '15 at 15:42
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Say $F=\mathbb{F}_q$ with $q$ the power of a prime. Of course the splitting field of $p_i$ over $F$ is $\mathbb{F}_{q^{m_i}}$. The splitting field of $f$ over $F$ is the smallest field extension of $F$ containing all the roots of $f$, that is, all the roots of all the $p_i$. Hence it is the smallest field containing $\mathbb{F}_{q^{m_i}}$ for all $i$. We conclude by noticing that $\mathbb{F}_{q^a}\subseteq \mathbb{F}_{q^b} \iff a\mid b$.

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protected by user26857 Sep 9 '15 at 21:49

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