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How can we show that the polynomial $f(x)=1+x+x^3+x^4$ is not irreducible over any field?

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It suffices to show that it's reducible over every prime field (so $\mathbb{F}_p$ for all $p$ and $\mathbb{Q}$). – Qiaochu Yuan Nov 15 '11 at 5:57
$f(x)$ seems irreducible over $\mathbb Q$. Am I missing something? – Srivatsan Nov 15 '11 at 6:07
maybe the question is for any finite field... – N. S. Nov 15 '11 at 6:24
There is a fix: it is not irreducible over some fields and not not irreducible over others. – André Nicolas Nov 15 '11 at 7:20
You can factor this polynomial as $1+x+x^3+x^4 = (1 + x)^2 (1 - x + x^2)$ – Aleks Vlasev Nov 15 '11 at 9:28
up vote 7 down vote accepted

-1 is an element of every field and is also a root of $f(x)$. Thus $x+1$ divides $f(x)$ and so $f(x)$ is not irreducible.

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