The affine coordinate ring of twisted cubic curve, $Y$ is $A(Y)=k[x,y,z]/(z-x^3, y-x^2)$? I am working on the following problem:
Let $Y \subset A^3$ be the set $Y={(t,t^2,t^3)|t\in k}$
($A^3$ is the affine 3-space over $k$ an algebraically closed field.)
Show that $Y$ is an affine variety of dimension 1. Find generators for the ideal $I(Y)$. Show that $A(Y)$ is isomorphic to a polynomial ring in one variable over $k$. 
I have found an answer which goes like this:
Clearly, $A(Y)=k[x,y,z]/(z-x^3, y-x^2)$. Note that $A(Y) =  
k[x,x^2,x^3]= k[x]$ so Y is irreducible, and $Y$ is an affine variety. and dim $Y$ = dim $A(Y)$ = dim $k[x]$ = 1 and $I(Y) = (z-x^3,y-x^2)$
I am finding it difficult to understand this answer, it doesn't even look like it has addressed each part of the answer. In particular, how does one show $A(Y)=k[x,y,z]/(z-x^3, y-x^2)$? 
I would also greatly appreciate a better way to approach this.
 A: In order to show that $A(Y) = k[x, y, z]/(z-x^3, y-x^2)$, the simplest way is to show that $Y = Z(z-x^3, y-x^2)$, i.e. the variety in $\Bbb A^3_k$ that has the coordinate ring $ k[x, y, z]/(z-x^3, y-x^2)$ has the same underlying set of points as the image of the map $t \mapsto(t, t^2, t^3)$.
This is done by showing inclusion both ways. First of all, it's easy to check that $Y \subseteq Z(z-x^3, y-x^2)$, since for any $t \in k$ we have $t^3-t^3 = 0$ and $t^2 - t^2 = 0$.
For the other inclusion, say we have a point $(a, b, c) \in Z(z-x^3, y-x^2)$. That means that $c - a^3= 0$ and $b - a^2 = 0$, or in other words, $b = a^2$ and $c = a^3$. Then clearly $a \mapsto (a, b, c)$ by the parametrisation that defines $Y$, so $(a, b, c) \in Y$.
As for how to guess at $(z-x^3, y-x^2)$ as the defining ideal in the first place, it's mostly practice. There are, however, a few pointers to get you started: Since $Y$ is a curve in $\Bbb A^3_k$, you should try to look for two generators as a start. Some times you might need more (like for instance $t \mapsto (t^3, t^4, t^5)$, which corresponds to the ideal $(x^5 - z^3, x^4 - y^3, y^5 - z^4)$). Other than that, you just stare at the parametrisation and write down any relations that you see fit, in as low degree as possible. Some times you write too many, some times the presence of one generator means you can simplify another.
