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Let $(P_n)_n$ be the sequence of polynomials on $\mathbb F_3$ defined by $$P_n=\prod_{\substack{h\in\mathbb F_3[t]\\\deg\,h=n\\h\text{ monic}}}h\qquad (P_0=1).$$ For every $r\in\mathbb N^*$, computations on Maple suggest that $\sum_{i=0}^{r-1}P_i$ is not divisible by any irreducible polynomial of $\mathbb F_3[t]$ with degree $r$.

Is it true in whole generality? If yes any hint or solution to prove that will be welcome. Thanks in advance

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    $\begingroup$ Can you please be more specific than "computations suggest" by telling us what range of $r$ you checked? And why are you telling us only for $\mathbf F_3$? Have you checked data for the same task over $\mathbf F_p$ for other primes $p$ besides $3$? Do you know that $P_n = \prod_{1 \leq m \leq n} (T^{3^m}-T)^{3^{n-m}}$ for $n \geq 1$ (and similarly over other $\mathbf F_p$ with $p$ replacing $3$ in the exponent)? $\endgroup$
    – KConrad
    May 1, 2015 at 18:32
  • $\begingroup$ I tried until r=7. For r=8, I stopped maple after 2 days . I did not receive any answer from maple. For p=5, reasonnable computations seems to be lesser than r=5 $\endgroup$
    – joaopa
    May 1, 2015 at 20:39
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    $\begingroup$ And what about $p = 2$? $\endgroup$
    – KConrad
    May 1, 2015 at 22:18

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