The polynomial $(X^2+2)^n+5(X^{2n-1}+10X^n+5)$ is irreducible in $\mathbb Z[X]$

Question

Prove that for any postive integer $n$, the polynomial $$(X^2+2)^n+5(X^{2n-1}+10X^n+5)$$ is irreducible in $\mathbb Z[X]$.

I have try use Eisenstein's criterion and can't it

You're right that Eisenstein's criterion won't help directly. The only divisor of the coefficient of $X^{2n-1}$ is 5 and $5$ does not divide the constant term, $25+2^n$. — Michael Burr, Mar 04 '15 at 15:51

score 2 · Answer 1 · edited Apr 13 '17 at 12:21

2

This is an immediate application of the Schönemann's Irreducibility Criterion. It also follows by reducing modulo $5$ twice.

edited Apr 13 '17 at 12:21

Community

answered Mar 10 '15 at 21:46

user26857

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What do you mean by "reducing mod 5 twice"? – azimut Mar 10 '15 at 22:06
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Reduce once, draw the conclusions, cancel $(X+2)^n$ and then $5$, reduce again. – user26857 Mar 10 '15 at 22:23

1 Answers1