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9.16. Let $F$ be a field of characteristic 0 , let $f(x)?F[x]$, and let $K/F$ be a splitting field for $f(x)$ over $F$. This exercise asks you to prove Proposition 9.34 , which states the $K$ is the splitting field of a separable polynomial in $F[x]$ (a) We know from Corollary 7.20 that we can factor $f(x)$ as a product of irreducible polynomials, say $f(x)=cg_{1}(x)_{e_{1}}g_{2}(x)_{e_{2}}?g_{r}(x)_{e_{r}}$ where $g_{1}(x),…,g_{r}(x)?F[x]$ are distinct monic irreducible polynomials. Prove that $g_{i}(x)$ and $g_{j}(x)$ have a common root $?i=j$. (b) Let $g(x)=g_{1}(x)g_{2}(x)?g_{r}(x)$. Prove that $g(x)$ is a separable polynomial. (c) Prove that $K$ is the splitting field of $g(x)$ over $F$.