Let $m>n>1$ be odd positive integers, show that $x^m+x^n+x+1$ is irreducible in $\mathbb{Z}[x]$.
I guess the proposition is true in $\mathbb{Z}_3$, but have absolutely no idea about how to prove it.
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1You can find a proof for the more general situation here https://www.emis.de/journals/INTEGERS/papers/g16/g16.pdf – Vlad Matei Mar 14 '22 at 07:43
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Great thanks, this article is really useful. – OrthoPole Mar 14 '22 at 08:04
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1It's not true in $\mathbb{Z_3}[x]$ in general, for example $$x^9+x^3+x+1=(x^3+x^2+x+2)(x^3+x^2+2x+1)(x^3+x^2+2)$$ – Sil Mar 16 '22 at 11:49