Prove that $p(x)=(x-1)(x-2) \cdots (x-n) + 1$ is irreducible over $\mathbb{Z}$ for all $n \geq 1$, $n \neq 4$.
I do not clearly see how to solve this problem and what is so special about the integer $4$ here?
Thanks for your help in advance
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1$(x-1)(x-2)(x-3)(x-4)+1=(x^2 - 5x + 5)^2$ is reducible for $n=4$. – Dietrich Burde Nov 19 '14 at 14:34
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@DietrichBurde That exception is included in the problem statement. – Caleb Stanford Nov 19 '14 at 14:37
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@Goos : this answers the question of the OP on why $n=4$ is special. user26857 also answered to this. – Dietrich Burde Nov 19 '14 at 14:39
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@DietrichBurde Okay. Though I think that the OP is not asking why it factors for $n=4$, but rather a deeper reason for why the proof of the proposition that it is irreducible (whatever that proof is) fails when $n=4$. – Caleb Stanford Nov 19 '14 at 14:41
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@DietrichBurde That is not very deep at all. If you read the answer on the linked duplicate question, you will see that the reason it fails is closer to the fact that $4 / 2 = 2$ is not larger than the difference $|1 - (-1)|$. – Caleb Stanford Nov 19 '14 at 14:47
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@Goos: Well, if you think this is deeper, then OK. – Dietrich Burde Nov 19 '14 at 14:48