(a) Let $X \subseteq \mathbf{A}^{n}$ be an affine algebraic variety defined over the field $k$.

Define the tangent space $T_{p} X$ for $p \in X$, and the dimension of $X$ in terms of $T_{p} X$.

Suppose that $k$ is an algebraically closed field with char $k>0$. Show directly from your definition that if $X=Z(f)$, where $f \in k\left[x_{1}, \ldots, x_{n}\right]$ is irreducible, then $\operatorname{dim} X=n-1$.

[Any form of the Nullstellensatz may be used if you state it clearly.]

(b) Suppose that char $k=0$, and let $W$ be the vector space of homogeneous polynomials of degree $d$ in 3 variables over $k$. Show that

$$U=\left\{(f, p) \in W \times k^{3} \mid Z(f-1) \text { is a smooth surface at } p\right\}$$

is a non-empty Zariski open subset of $W \times k^{3}$.