6
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Figured it out, in case anyone is working on it. $\endgroup$ – Travis62 Dec 21 '18 at 19:38
  • 1
    $\begingroup$ 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
3
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.