# Show the following polynomial is Irreducible over the given ring

Studying for a qualifying exam and this practice problem flat out has me stumped. I wish to show that the polynomial $$(y+8)^2x^3 - x^2 + (y+7)(y+8) - y - 12$$ is irreducible over $$\mathbb{Q}[x, y]$$. My thought was to use Eisenstein's for $$\mathbb{Q}[x][y]$$ and $$\mathbb{Q}[y][x]$$, however both variations haven't yielded a solution. For example, in $$\mathbb{Q}[y][x]$$, the prime ideal would need to contain $$-1$$, but this cannot happen. Writing this polynomial as a polynomial in $$\mathbb{Q}[x][y]$$ yields the polynomial $$(x^3+1)y^2 + (16x^3 + 14)y + 64x^3 - x^2 + 44$$, but haven't been able to find a prime ideal to satisfy Eisenstein's. Any idea would be appreciated.

• Figured it out, in case anyone is working on it. – Travis62 Dec 21 '18 at 19:38
• Might I suggest that you post an answer to your own question? This would remove it from the "unanswered" tab. – Pierre-Guy Plamondon Dec 23 '18 at 16:29

Consider the homomorphism $$y \to -7$$ from $$\mathbb{Q}[x, y] \to \mathbb{Q}[x]$$. Then the image is a proper prime ideal of $$\mathbb{Q}[x]$$ and one can show then that the preimage must also be a prime ideal in $$\mathbb{Q}[x, y]$$.