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Problem. Check whether the following ideals are maximal or prime in $\mathbb{Z}[X_1,X_2]$ and $\mathbb{Q}[X_1,X_2]$:
i) $(X_1,X_2)$
ii) $(X_1+X_2)$
iii) $(X_1,X_2,2)$
iv) $(X_1+X_2^2,X_1^2+X_2)$

I know the definitions of these and I know how these ideals look like. I also know that "modding out" a prime ideal yields an integral domain, and a maximal ideal yields a field. I also know that maximal implies prime. Is there a general approach to this problem?

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    $\begingroup$ Modding out by a prime ideal yields an integral domain (not a field), and modding out by a maximal ideal yields a field. $\endgroup$ – tylerc0816 Dec 7 '13 at 14:53
  • $\begingroup$ Yes, you are right, I confused things in my head. $\endgroup$ – blst Dec 7 '13 at 14:58
  • $\begingroup$ Not exactly my meat, but it seems to me that you have to use the method of ingenious devices, mostly. In some cases, you can easily find what the quotient looks like (note: not “how” it looks like). In the last case, I’d draw the associated picture, noticing that $y=-x^2$ and $x=-y^2$ intersect in two separate points, so you can find functions that vanish at one and not the other. $\endgroup$ – Lubin Dec 7 '13 at 15:52
  • $\begingroup$ I don't see how this helps me here :( $\endgroup$ – blst Dec 7 '13 at 16:14
  • $\begingroup$ why don't you just try with whatever you know? what do you think will $\mathbb{Z}[X_1,X_2]/(X_1,X_2)$ be? $\endgroup$ – user87543 Dec 7 '13 at 16:26

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