Irreducibles Over A Field Of Characteristic $p$ May Have Repeated Zeros

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Problem 83 (Irreducibles Over A Field Of Characteristic $p$ May Have Repeated Zeros)

Let $F$ be a field of characteristic $p$ 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 $a_1, \ldots, a_n\in E$.

1. Show that $f(x)$ might have a zero with multiplicity greater than one (give an example of such a polynomial).
2. If $f(x)$ has a multiple zero, then what can we say about the coefficients of $f(x)$?

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