Exercise 4, § 3

(a) Show that the ring $A = \mathbf{Q}[X, Y]/p$, where $p$ is the principal ideal generated by $X^2 + Y^2 - 1$ in $\mathbf{Q}[X, Y]$, is a non-factorial Krull domain (if $x$ is the image of $X$ in $A$, show that $x$ is an extremal element of $A$ but that $Ax$ is not a maximal ideal of $A$).

(b) Show that the ring $A \otimes_{\mathbf{Q}} \mathbf{Q}(i)$ is factorial (prove that this ring is isomorphic to the quotient of $\mathbf{Q}(i)[X, Y]$ by the principal ideal $(XY - 1)$; compare with Exercise 3 (b)).