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I have the following past exam paper question, a similar sort of question seems to come up every year.. And I'm completely lost with it...

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $(y^2-xy-2zx, y^3+z^2+1, x^2yz-zy)$.

Show that $J$ is zero-dimensional.

What is the dimension, as a vector space over $\mathbb{Q}$, of $\mathbb{Q}[x,y,z]/J$?

Define the ideal $I := (xy^2+2xz-yz, x^2yz+y^2)$. Explain why $I$ is not zero-dimensional.

I know the definition for an ideal to be zero-dimensional, but I cant find an example of how to use to it to show an ideal is zero-dimensional...

Defintion. An ideal $I\subset K[X_1,...,X_n]$ is zero-dimensional if the $K$-vector space $K[X_1,...,X_n]/I$ is finite dimensional.

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  • $\begingroup$ This sounds like a homework problem. I voted to close. $\endgroup$ – Angelo May 4 '13 at 12:08
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Your question is better suited for math.stackexchange.com and will probably be closed here quickly (don't get discouraged by this -- MathOverflow is exclusively for research level mathematics, while math.stackexchange.com is a great resource for questions of any level).

But, in the mean time: Do you know how to form a Groebner basis? This is an easy way to prove that a finitely presented ideal is zero-dimensional (and is easy to do by hand in this example with a lexicographic monomial ordering $\prec$ s.t. $z \prec y \prec x$).

Think about what it means for $J$ to be zero-dimensional: $J$ is zero-dimensional iff $V_{\mathbb{C}}(J)$ contains only finitely many points. For this to happen, a (monic) Groebner basis for $J$ must contain polynomials $p_1, p_2, p_3$ s.t. $LM(p_1)$ = $x^{k_1}$, $LM(p_2)$ = $y^{k_2}$, and $LM(p_3)$ = $z^{k_3}$, where $LM(p_i)$ is the leading monomial of $p_i$, and $k_i \in \mathbb{N}^{+}$.

In general, to decide whether or not an ideal $I \subset \mathbb{Q}[x_1, \ldots, x_n]$ is zero-dimensional, it suffices to compute a (monic) Groebner basis for $I$ and verify that it contains, for each indeterminate $x_i$, a polynomial $p_i$ s.t. $LM(p_i) = x_i^{k_i}$ for some $k_i \in \mathbb{N}^{+}$.

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  • $\begingroup$ Sorry, I didn't realise this was for research level.. I have actually tried asking on math.stackexchange.com but with very little response, only someone quoting me the definiton. I am still trying to compute the Groebner basis for this, but I don't really understand what I have to do to with the Groebner basis to show its zero dimensional. Ive tried looking everywhere for an example of this, but cant find one anywhere.. Thank you for your help in the right direction though. $\endgroup$ – MathStudent21 May 4 '13 at 12:17

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