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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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    You can factor this polynomial as $1+x+x^3+x^4 = (1 + x)^2 (1 - x + x^2)$2011-11-15

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-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.