How can we show that the polynomial $f(x)=1+x+x^3+x^4$ is not irreducible over any field?
$1+x+x^3+x^4$ is not irreducible for any field
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abstract-algebra
polynomials
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2It suffices to show that it's reducible over every prime field (so $\mathbb{F}_p$ for all $p$ and $\mathbb{Q}$). – 2011-11-15
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3[$f(x)$ seems irreducible over $\mathbb Q$.](http://www.wolframalpha.com/input/?i=is%201%2Bx%5E2%2Bx%5E3%2Bx%5E4%20irreducible%3F&lk=2) Am I missing something? – 2011-11-15
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2maybe the question is for any finite field... – 2011-11-15
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0I agree it's irreducible over $\mathbb{Q}$ and also find it's irreducible mod 5 so I guess that's not the fix either? – 2011-11-15
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5There is a fix: it is not irreducible over some fields and not not irreducible over others. – 2011-11-15
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0I'm sorry. I did a mistake in writing the wrong polynomial. I've fixed it now. – 2011-11-15
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4You can factor this polynomial as $1+x+x^3+x^4 = (1 + x)^2 (1 - x + x^2)$ – 2011-11-15
1 Answers
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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.