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Prove that $x^{6}+30x^{5}-15x^{3}+6x-120$ can't be written as a product of two polynomials of rational coefficients and positive degrees.

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Eisenstein polynomial. – Cocopuffs Dec 12 '12 at 15:07
In that case we say the polynomial is irreducible over the complex numbers. That will give you something to look up. – GEdgar Dec 12 '12 at 15:08
@GEdgar over the complex numbers? – David Mitra Dec 12 '12 at 15:09
Look at $p=3$ and use Eisenstein. – PAD Dec 12 '12 at 15:11
The question says rational coefficients. If the OP does not use the word "irreducible" then he/she likely does not know what "Eisenstein" means. – GEdgar Dec 12 '12 at 15:13
up vote 4 down vote accepted

Suppose $f(x) = x^{6}+30x^{5}-15x^{3}+6x-120$ is not irreducible in $\mathbb{Q}[x]$. Then it is not irreducible in $\mathbb{Z}[x]$ by Gauss's lemma. However it is irreducible in $\mathbb{Z}[x]$ by Eisenstein's criterion using the prime number $3$. This is a contradiction. Hence $f(x)$ is irreducible in $\mathbb{Q}[x]$

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