Proving certain polynomials are irreducible over $\mathbb{Z}$

Asked Oct 07 '18 at 21:03

Active Oct 07 '18 at 21:03

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Show that the polynomial $(x-1)(x-2)...(x-n)-1$ is irreducible over $\mathbb{Z}$. [hint: if the polynomial factors consider the values of the factors at $x = 1,2,3....,n$.

Show that the polynomial $(x-1)(x-2)...(x-n)+1$ is irreducible over $\mathbb{Z}$ $\forall n \geq 1, n \neq 4$.

I've been running around in circles on this one for awhile now and can't seem to get anywhere! Some insights would be greatly appreciated! Thanks yall!

asked Oct 07 '18 at 21:03

Math is hard

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@lulu I think the referenced Q does not consider the $+1$ variant. – Hagen von Eitzen Oct 07 '18 at 21:10
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@HagenvonEitzen The question cited as a duplicate for that one does – lulu Oct 07 '18 at 21:12
@lulu Voting with different duplicates will show all of them. – egreg Oct 07 '18 at 21:40
@egreg Didn't know that, thanks! – lulu Oct 07 '18 at 22:04
@lulu I'm not sure who can, but I see also an “edit” link below the list of duplicates. It's possibly a “gold badge privilege”. – egreg Oct 07 '18 at 22:06
@egreg Next time it comes up, I'll give it a try. – lulu Oct 07 '18 at 22:12

Proving certain polynomials are irreducible over $\mathbb{Z}$

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