Show that some set is an algebraic set

Question

Show that $\{(t,t^2,t^3):t\in k\}$ for a field $k$ is an algebraic set.

Just by looking at the points, i see that they are zeros of $F_1(x,y,z)=xy-z$, $F_2(x,y,z)=xz-y^2$ and $F_3(x,y,z)=yz-x^5$.

I guess Zero set of ideal $I=(xy-z,xz-y^2,yz-x^5)$ is $\{(t,t^2,t^3):t\in k\}$..

could not prove this...

Answer 1

There are even lower degree polynomials that become zero for $x=t, y=t^2, z=t^3$. For example, $y-x^2$ and $z-x^3$. The set of $(x, y, z)$ that satisfy the polynomial equations $y=x^2$ and $z=x^3$ is an algebraic set (by definition). Show that the set of all $(t, t^2, t^3)$ is equal (pointwise, as sets) to this algebraic set. One inclusion is obvious, but the other is also almost equally obvious.

Note, it is important to "eyeball" the right polynomials, in this case $y-x^2$ and $z-x^3$; the more polynomials you put in your ideal, and the higher the degree, the harder will be your job to prove the equality of the two sets.

Answer 2

You know that if a point can be written in the form $(t,t^2,t^3)$, then it satisfies the equations of your ideal. Now you have a point $(x,y,z)$ in your ideal, and you want to show that it can be written in the form $(t,t^2,t^3)$. What should $t$ be? Do your conditions on $x$, $y$, and $z$ guarantee that this choice of $t$ will work?