Degree of splitting field of polynomial over a finite field

Question

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 $n_1,...,n_k$?

Answer 1

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 $\mathbb{F}_{q^a}\subseteq \mathbb{F}_{q^b} \iff a\mid b$.