Irreducibles Over A Field Of Characteristic Zero Have No Repeated Zeros

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Problem 82 (Irreducibles Over A Field Of Characteristic Zero Have No Repeated Zeros)

Let $F$ be a field of characteristic zero and $f(x)\in F[x]$ be irreducible over $F$. Suppose that $E$ is a splitting field for $f(x)$, so that we can write $f(x) = c(x-a_1)(x-a_2)\cdots(x-a_n)$ for $c\in F$ and $a_1, \ldots, a_n\in E$. Show that $a_i\neq a_j$ if $i\neq j$, in other words show that $f$ has no zero of multiplicity greater than one.

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