I got $$P(x)=1+\prod_{i=0}^{2021} (x-i)$$ and need to use Eisensteins's Criteria to solve the irreducibility of $P(x)$ but I found a problem how to elaborate the coefficient and choosing prime $p$. Anyone can help?
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1Maybe I got it wrong but the leading and trailing coefficients both are equal to 1 since your product start with i=0. So Eisenstein's criterion does not apply to P(x). Maybe try P(x+n) for some n? – Arnaud May 18 '22 at 13:05
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1This does not work in general : $x(x-1)(x-2)(x-3)+1$ is reducible ! – Peter May 18 '22 at 13:42
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For this particular polynomial , there might be a possibility to apply Eisenstein – Peter May 18 '22 at 13:45
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1As to the question of what the expansion of $x(x-1)(x-2)\cdots (x-k)$ is... that is precisely what the Stirling Numbers of the First Kind are defined as being. – JMoravitz May 18 '22 at 14:04
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$\prod _{n=0}^k (x-n)=\frac{\Gamma (x+1)}{\Gamma (x-k)}$ – Steven Clark May 18 '22 at 14:24
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Is there perhaps a square missing? See my "answer". – Dietrich Burde May 18 '22 at 18:15
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The usual exercise is to show that $$ P(x)=1+\Pi_{i=0}^{2022} (x-i)^2 $$ is irreducible over $\Bbb Q$. This has been solved here:
$[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$
Dietrich Burde
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