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I'm hoping someone can give me a nudge in the right direction...

Let $F$ be a finite field, and let $f(x)$ be a nonconstant polynomial whose derivative is the zero polynomial. Prove that $f$ cannot be irreducible over $F$.

I've got that every root of $f$ is a multiple root and that for $F=\mathbb{F}_{p^r}$, the exponent of every term of $f$ is a multiple of $p$.

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up vote 4 down vote accepted

Hint $\ $ prime $\rm\:P\equiv 0,\ \ A^P\equiv A,\ B^P\equiv B\ \ \Rightarrow\ \ A\:X^{JP} +\!\: B\:X^{KP}\equiv\: (A\:X^J +\!\: B\:X^K)^P$

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Hint: there is a polynomial $g$ over $F$ such that $f=g^p$. Do you see what it is?

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2  
+1 Also, all the coefficients are $p^\text{th}$ powers. Hopefully the OP knows why that is true? – Jyrki Lahtonen Apr 2 '12 at 20:03

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