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.

  • $\begingroup$ Figured it out, in case anyone is working on it. $\endgroup$ – Travis62 Dec 21 '18 at 19:38
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    $\begingroup$ Might I suggest that you post an answer to your own question? This would remove it from the "unanswered" tab. $\endgroup$ – 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 $\mathbb{Q}[x, y]$.


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