Exercise 23, § 1

(a) Let $A$ be a pseudo-Bezoutiandomain (Exercise 21) and $K$ its field of fractions; for every polynomial $f \in A[X]$ a g.c.d. of the coefficients off (determined to within an invertible element of $A$) is called a content off. Let $f, g$ be two polynomials of $A[X]$; show that for $f$ to divide $g$ in $A[X]$, it is necessary and sufficient that $f$ divide $g$ in $K[X]$ and that a content off divide a content of $g$ (use Exercise 12) (cf. Exercise 30 (c)).

(b) Deduce from (a) that, if $A$ is a pseudo-Bezoutian (resp. pseudo-principal) domain, then so is $A[X]$. Moreover, if (a_i) is a finite (resp. any) family of elements $\neq 0$ of $A$ and d a g.c.d. of this family in $A$, d is also a g.c.d. of this family in $A[X]$.

(c) Deduce from (b) that, if $A$ is pseudo-Bezoutian (resp. pseudo-principal) then $A[X_\lambda]_{\lambda \in L}$ is pseudo-Bezoutian (resp. pseudo-principal) for every family $(X_\lambda)_{\lambda \in L}$ of indeterminates.