Homework Help: Prove 1+8x-12x^3+2x^4 irreducible over Q[x]

1. Dec 9, 2011

1. The problem statement, all variables and given/known data
Determine if 1+8x-12x^3+2x^4 irreducible over Q[x]

2. Relevant equations
Gauss's Lemma, Eisenstein Criterion

3. The attempt at a solution

If we multiply g(x)=1+8x-12x^3+2x^4 by 2^3, and then make the substitution y=(2*x), we recover a monic polynomial: h(y) = 8+32y-12y^3+y^4, but Eisenstein does not apply. h(y+/-1) doesn't help either. h(y) is of fourth degree, so it doesn't even suffice to check for absence of roots. But, I do know that if irreducibility in Z is shown, irreducibility over Q follows, for suppose such an f(x) is reducible over Q. Then, by Gauss's Lemma, there exists a nontrivial factorization of f(x) over Z into monic polynomials, but if f(x) is irreducible over Z, this is a contradiction. Thus, f(x) must be irreducible over Q.

Still, I have tried many things, so any help is appereciated.

2. Dec 9, 2011

Dick

Are you sure h(y+/-1) doesn't satisfy Eisenstein's criterion? It sure looks like they do to me.

3. Dec 10, 2011