Exercise 21, § 2

(a) Let $\mathbf{A}$ be an integral domain, $f, g$ two polynomials in $\mathbf{A}[X]$ and $h = fg$. Let $a, b, c$ denote the ideals of $\mathbf{A}$ generated respectively by the coefficients of $f, g, h$. Show that, if $\deg(g) = n$, then $a^{n+1}b = a^n c$. (Let $f(X) = \sum_{i=1}^m a_i X^i, g(X) = \sum_{j=1}^n b_j X^j$; for every increasing sequence

of integers $\leq m$ and all $j$, let take on the set of $u_{\sigma,j}$ a total ordering such that, for $j < j'$, $u_{\sigma,j} < u_{\tau,j'}$ for all $\sigma, \tau$, and, for all $j$, $u_{\sigma,j} \leq u_{\tau,j}$ if and only if $\sigma \leq \tau$ in the lexicographical ordering on $[0, m]^{n+1}$. Then argue by induction on this totally ordered set.)

(b) Deduce from (a) that, if $\mathbf{A}$ is Pruferian (Exercise 12), then $c = ab$.

(c) Taking $\mathbf{A}$ to be the polynomial ring $\mathbf{Z}[Y]$, give an example of two polynomials $f, g$ of the first degree in $\mathbf{A}[X]$ such that $c \neq ab$.