I am trying to work through Hartshorne's book and while working through one of the exercises, I need to show the following:
Let $k$ be an algebraically closed field. Let $Y \subseteq A^3$ be the set $Y=\{(t,t^2,t^3)\;|\;t \in k\}$. Show that $Y$ is an affine variety.
The rest of the problem I have completed. I'm just not sure how to show that this set is closed and irreducible. I assume that they intend for $A^3$ to have the Zariski topology.
Hint: First show that $Y$ is the zero set of the ideal $I = (z - x^3, y - x^2)$. Then show that $I$ is a prime ideal by showing that the quotient $k[x, y, z]/I$ is isomorphic to $k[x]$ (and hence an integral domain).