Exercise 5, § 3

a) Let $k$ be a noetherian factorial ring, $B$ the polynomial ring $k[X_1, \ldots, X_n]$ with $n \geq 3$; let $g_i$ ($0 \leq i \leq r$) be elements of $k[X_3, \ldots, X_n]$ where $g_0$ is extremal. Put $g = X_1X_2 - \sum_{i=0}^r g_i X_1^i$, and consider the quotient ring $A = B/gB$. Now prove that $A$ is factorial. (Let $S$ be the multiplicative subset of $A$ generated by 1 and the image of $X_1$ in $A$; apply to $S^{-1}A$ Prop. 3 of No. 4).

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

c) If $F = X_1X_2 - X_3X_4$, now prove that the ring $k[X_1, X_2, X_3, X_4]/(F)$ is not factorial. (Show that the images of the $X_i$ in this ring are extremal elements).