(a) Let $k$ be an algebraically closed field of characteristic 0 . Consider the algebraic variety $V \subset \mathbb{A}^{3}$ defined over $k$ by the polynomials

$$x y, \quad y^{2}-z^{3}+x z, \quad \text { and } x(x+y+2 z+1)$$

Determine

(i) the irreducible components of $V$,

(ii) the tangent space at each point of $V$,

(iii) for each irreducible component, the smooth points of that component, and

(iv) the dimensions of the irreducible components.

(b) Let $L \supseteq K$ be a finite extension of fields, and $\operatorname{dim}_{K} L=n$. Identify $L$ with $\mathbb{A}^{n}$ over $K$ and show that

$$U=\{\alpha \in L \mid K[\alpha]=L\}$$

is the complement in $\mathbb{A}^{n}$ of the vanishing set of some polynomial. [You need not show that $U$ is non-empty. You may assume that $K[\alpha]=L$ if and only if $1, \alpha, \ldots, \alpha^{n-1}$ form a basis of $L$ over $K$.]