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Prove that the polynomial $3x^5-12x^4+21x^3-9x+2$ is irreducible over the rationals

$p=2: f(x)=x^5+x^3+x$..this has root..So this is not irreducible over $Z_2$ so Can I conclude $f(x)$ is irreducible over the rationals?

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Why would you expect that demonstrating a polynomial is reducible in one context would imply that it is irreducible in a different context? – Zev Chonoles Mar 29 '13 at 9:32

Try Eisenstein's criterion on $x^5 P(1/x)$.

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Ooh, clever! +1 – Zev Chonoles Mar 29 '13 at 9:40

Hint: Try $p=3$ instead. Remember to appeal to Gauss's lemma at some point.

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Using Eisntein's irreducibility Criterion P must not divide $a_n$ and $P^2$ must not divide $a_0$ for $f(x)$ to be irreducible over the field of rational numbers but $a_n=3$ which is divisible by 3. I am trying to figure this out so please do let me know if I am wrong. – Kj Tada Apr 4 '13 at 10:41

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