How to prove that $3x^5-12x^4+21x^3-9x+2$ is irreducible over the rationals?

Question

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?

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 09:32

score 13 · Answer 1 · answered Mar 29 '13 at 09:39

13

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

answered Mar 29 '13 at 09:39

Robert Israel

score 1 · Answer 2 · answered Mar 29 '13 at 09:39

1

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

answered Mar 29 '13 at 09:39

Zev Chonoles

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 04 '13 at 10:41

2 Answers2