Exercise 5, § 2

(a) Let $k$ be a field, $K$ the field $k(X, Y)$ of rational functions in two indeterminates over $k$ and $A$ the polynomial ring $K[Z]$ in one indeterminate, which is a principal ideal domain. Let $L$ be the subfield $k(Z, X + YZ)$ of $k(X, Y, Z)$; show that $A \cap L$ is not a Dedekind domain (prove that there are prime ideals which are not maximal and $\neq 0$ in this ring).

(b) Let $A$ be a Dedekind domain which is not a principal ideal domain and for which the ideal class group $C(A)$ is finite \* (the ring of integers of a finite algebraic extension of $\mathbf{Q}$ has the latter property). \* Let $(a_i)_{i=1}^s$ be a representative system of the group $I(A)$ of fractional ideals $\neq 0$ modulo the subgroup of fractional principal ideals, consisting of integral ideals of $A$ and let $p_i$ ($1 \leq i \leq s$) be the prime ideals dividing at least one of the $a_i$. Let $S$ (resp. $T$) denote the multiplicative set of elements $x \in \mathbf{A}$ such that $v_{p_i}(x) = 0$ for all $i$ (resp. consisting of 1 and the $x \in \mathbf{A}$ such that $v_q(x) = 0$ for all the prime ideals $q$ distinct from the $p_i$ and $v_{p_i}(x) \geq 1$ for all $i$; the hypothesis implies that $T$ is not reduced to 1). Show that $S^{-1}\mathbf{A}$ and $T^{-1}\mathbf{A}$ are principal ideal domains and that $\mathbf{A} = (S^{-1}\mathbf{A}) \cap (T^{-1}\mathbf{A})$.