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Let $p$ be prime. Consider monic polynomials of degree $d$ over $\mathbb Z / p \mathbb Z$. Denote the number of such polynomials with degree less than $p$, which are not zero for all $x \in \mathbb Z / p \mathbb Z$, with $m_d$. Let $d \geq p$. Then I have to show that there are $m_p p^{d-p}$ of such polynomials which are not zero for all $x \in \mathbb Z / p \mathbb Z$.

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I am confused. You appear to define $m_{d}$ as the number of monic polynomials of degree $d$ with coefficients in $\mathbb Z / p \mathbb Z$ which have no roots in $\mathbb Z / p \mathbb Z$. But I don't understand which polynomials you want to count then. –  Andreas Caranti Feb 8 '13 at 16:51
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It seems you want to prove $m_d=m_pp^{d-p}$? If so, I think it would be clearer if you put it that way. It's confusing that you say "such polynomials" but then only repeat one of the conditions on these polynomials from the previous sentence, so it's not clear whether these are the same kind of polynomials as in the other sentence or not. Also it's not clear whether the polynomials should be non-zero for all $x$ or whether it shouldn't be the case that they're not zero for all $x$. –  joriki Feb 8 '13 at 18:23

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Hint: The polynomials that are zero for all $x \in \mathbb F_p=\mathbb Z/p\mathbb Z$ are precisely those in the ideal generated by $x^p-x$.

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