Let $k$ be an algebraically closed field of characteristic zero. Prove that an affine variety $V \subset \mathbb{A}_{k}^{n}$ is irreducible if and only if the associated ideal $I(V)$ of polynomials that vanish on $V$ is prime.

Prove that the variety $\mathbb{V}\left(y^{2}-x^{3}\right) \subset \mathbb{A}_{k}^{2}$ is irreducible.

State what it means for an affine variety over $k$ to be smooth and determine whether or not $\mathbb{V}\left(y^{2}-x^{3}\right)$ is smooth.