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I took my friend's notes and there was a question which was asking if $$x^5+9x^4+12x^2+6$$ is irreducible in $ \Bbb Q[x]$. answer was "yes, because it is irreducible in $Z_3$[x]" But isnt $0\in Z_3$ a root of this polynomial? I think it is irreducible because Eisenstein's. (which is for p=3) am I wrong?

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up vote 5 down vote accepted

You are right, Eisenstein does it. And the polynomial is not irreducible over $\mathbb{Z}_3[x]$, it factors for example as $x\cdot x^4$.

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