Showing the following subset is an affine variety

Question

Let $k$ be an algebraically closed field and $$S= \{(t,t^2,t^3) \in \Bbb A^3(k)\}$$ where the symbols have their usual meanings. Then $S$ is the zero set of $$\{Y-X^2,Z-X^3\} \subset k[X,Y,Z]$$. I want to show that $S$ is an affine variety(irreducible algebraic set). This reduces to showing that the ideal generated by the above polynomials is prime which I'm not able to show. How does one proceed for such problems on general?

Answer 1

We have $I=(Y-X^2,Z-X^3)$ is the kernel of the (surjective) ring homomorphism $k[X,Y,Z]\to k[X]$ sending $X\mapsto X$, $Y\mapsto X^2, Z\mapsto X^3$. Since $k[X]$ is a domain, the ideal $I$ is prime.