(Solved): 9.16. Let F be a field of characteristic 0 , let f(x)F[x], and let K/F be a splitting field for ...

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 $$f(x)=c{g}_1(x{)}^{e_1}{g}_2(x{)}^{e_2}?g_r(x{)}^{e_r}$$ where $$g_1(x),\dots ,g_r(x)?F[x]$$ 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 $$F$$.