Exercise 31, § 1

(a) Let $A$ be a regularly integrally closed domain (Exercise 30) and $K$ its field of fractions; for every polynomial $P \in K[X]$, let $c(P)$ denote the divisor $\operatorname{div}(a)$, where $a$ is the fractional ideal of $K$ generated by the coefficients of $P$. Let $B$ denote the subring of $K(X)$ consisting of the rational functions $P/Q$ such that $c(P) \geq c(Q)$ (show that this condition depends only on the element $P/Q$, using Exercise 30 (a) and 12 (c)). Then $B \cap K = A$.

(b) Show that $B$ is a Bezoutian domain (Exercise 20) (observe that, if $P, Q$ are two polynomials of $A[X]$, $P + X^n Q$ divides $P$ and $Q$ in $B$ if $m$ is sufficiently large).

(c) Show that, if $A$ is a Krull domain, $B$ is a principal ideal domain (consider an increasing sequence of finitely generated fractional principal ideals of $B$). Give an example where $A$ is completely integrally closed but $B$ is not a principal ideal domain (cf. Exercise 22).

(d) Deduce from (c) an example of a Noetherian domain $B$ for which there exists a subfield $K$ of its field of fractions such that $B \cap K$ is not Noetherian.