We've been given the set $X = \{(t^3,t^4,t^5) \in \mathbb{A}^3 \mid t \in \mathbb{A}^1\}$ (where the underlying field $\mathbb{K}$ is infinite), and have been asked to show that $X = \mathbb{V}(J)$ where $J = \langle xz -y^2, x^3 - yz, z^2 - x^2 y \rangle$, which I have managed to prove. However, it then asks us to show that $\mathbb{I}(X) = J$ i.e. $\mathbb{I(V}(J)) = J$, and I'm unsure on how to approach it. One inclusion is true for any $J$, so it remains to show $\mathbb{I(V}(J)) \subseteq J$. I've tried writing any polynomial in $\mathbb{K}[x,y,z]$ in the form $f = f_1 (xz-y^2) + f_2 (x^3 -yz) + f_3 (z^2 - x^2 y) + g$, aiming to show that if $f \in \mathbb{I(V}(J))$, then $g = 0$, but haven't managed to get anywhere.
Is this the right way to approach it?
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$\begingroup$Your general approach is sensible, but you may need to be more systematic.
The last relation lets you eliminate all powers of $z$ beyond the first. So working modulo $J$, any coset has a representative of the form $f_1(x,y) + f_2(x,y) z$. Now can you use the first two relations to simplify $f_2$ any further?
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