Exercise 6, § 3

(a) Let $K$ be an algebraically closed field of characteristic $\neq 2$. Determine the graded prime ideals of the ring

(b) Let $k$ be an ordered field, $a, b, c$ elements $> 0$ of $k$ and $A$ the ring $k[X, Y, Z]/(aX^2 + bY^2 + cZ^2)$. Show that $A$ is a factorial domain. (Reduce it to the case where $k$ is a maximal ordered field, using Exercise 3 (b)); then prove, using (a), that every graded prime ideal of $A$ of height 1 is principal and apply Exercise 3 (a).)

(c) Let $k$ be an ordered field, $a, b$ elements $> 0$ of $k$ and $B$ the ring

Show that $B$ is a factorial domain (use (b) and Exercise 17 (a) of § 1.)

(d) Show that the domain $C = \mathbf{Q}[X, Y]/(X^2 + 2Y^2 + 1)$ is factorial, but that the domain $C \otimes_{\mathbf{Q}} \mathbf{Q}(i)$ (where $i^2 = -1$) is not factorial.