Show the following polynomial is Irreducible over the given ring

Question

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.

Answer 1

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 $\mathbb{Q}[x, y]$.