Exercise 5, § 3

(a) Let $k$ be a Noetherian factorial domain and $B$ the polynomial ring $k[X_1, \ldots, X_n]$ where $n \geq 3$; let $g_i$ ($0 \leq i \leq r$) be elements of $k[X_3, \ldots, X_n]$ where $g_0$ is extremal. Writing $g = X_1 X_2 - \sum_{i=1}^r g_i X_1^i$, consider the quotient ring $A = B/gB$. Show that $A$ is factorial. (Let $S$ be the multiplicative subset of $A$ generated by 1 and the image of $X$, in $A$; apply Proposition 3 of no. 4 to $S^{-1}A$.)

(b) Let $k$ be a field of characteristic $\neq 2$ and $F$ a homogeneous polynomial of the second degree in $k[X_1, \ldots, X_n]$ where $n \geq 5$, such that the corresponding polynomial function on $k^n$ is a non-degenerate quadratic form. Show that the domain $k[X_1, \ldots, X_n]/(F)$ is factorial. (Prove first that a homogeneous polynomial $G$ of the second degree in $k[X_1, \ldots, X_n]$, such that the corresponding polynomial function is a non-degenerate quadratic form, is extremal for $n \geq 3$. Then prove the proposition when $k$ is algebraically closed, using (a); finally pass to the general case with the aid of Exercise 3 (b).)

(c) If $F = X_1 X_2 - X_3 X_4$, show that the domain $k[X_1, X_2, X_3, X_4]/(F)$ is not factorial. (Show that the images of the $X$, in this ring are extremal elements.)