Exercise 1, § 4

Let $f$ be a polynomial in $n$ complex variables, with complex coefficients and not identically zero. Show that the complement in $\mathbf{C}^n$ of the set $S$ of points $z = (z_i)$ such that $f(z_1, z_2, \ldots, z_n) = 0$ (the "algebraic variety" with equation $f = 0$) is connected. (If $a, b$ are two distinct points of $\mathbf{C}S$, consider the intersections of $S$ and the complex line through $a$ and $b$.)