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Problem 74 (Existence Of A Splitting Field)
Let $F$ be a field and $f(x)\in F[x]$ have positive degree. We have already shown that there exists an extension field $E$ of $F$ in which $f(x)$ has a zero $a$, in other words we can write $f(x) = (x-a)g(x)$ where $a,g(x)\in E[x]$ (by the Factor Theorem). Prove that there exists an extension field $E'$ of $F$ in which $f(x)$ can be written as a product of linear factors, i.e. we can write $f(x) = (x-a_1)(x-a_2)\cdots (x-a_n)g$ where $a_1,a_2,\ldots, a_n,g\in E'$ and also $g$ is a unit.
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