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 $k[X, Y, Z]/(aX^2 + bY^2 + cZ^2)$. Now prove that A is a factorial ring. (Reduce to the case where $k$ is a maximal ordered field, using Exercise 3 b); proving then, using $a$), that every graded prime ideal of height 1 in A is principal, and apply Exercise 3 a)). c) Let $k$ be an ordered field, $a, b$ elements $> 0$ of $k$ and B the ring $k[X, Y]/(aX^2 + bY^2 + 1)$. Now prove that B is a factorial ring (use b) and Exercise 17 a) of § 1). d) Now prove that the ring $C = \mathbf{Q}[X, Y]/(X^2 + 2Y^2 + 1)$ is factorial, but that the ring $C' \otimes_{\mathbf{Q}} \mathbf{Q}(i)$ (with $i^2 = -1$) is not factorial.