I have some difficulty with the following problem:
Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an affine algebraic set and calculate $I(C)$. Show that the affine algebra $Γ(C)=k[X,Y,Z]/I(C)$ is isomorphic to the ring of polynomials $k[T]$.
I found a solution in the case when $k$ is an infinite field:
Clearly $C$ is the affine algebraic set $C=\big
Now since the function $f$ is an isomorfism between $C$ and $k$ follows that $\Gamma(C)\cong \Gamma(k)=k[T]$.
I can not find the ideal $I(C)$ if the field $k$ is finite.