Exercise 9, § 1

(a) Let $B$ be a discrete valuation ring, $A = B[[X]]$ the ring of formal power series in one indeterminate over $B$ and $C = A$, the ring of fractions $S^{-1}A$, where $S$ is the multiplicative set of $X^h \ (h \geq 0)$. Let $v$ be a normed valuation on $B$; for every element $f = \sum_{n \geq h} b_n X^n \neq 0$ of $C$ with $b_h \neq 0$ in $B$ ($h$ positive or negative), we write $s(f) = v(b_h)$. Show that $s$ is a Euclidean stathm on $C$ (Algebra, Chapter VII, § 1, Exercise 7), such that $s(fg) = s(f) + s(g)$ for non-zero $f, g$ in $C$. (If $s(f) = p \geq s(g) = q$ and $h$ is the least of the degrees of the terms $\neq 0$ in $f$, show that there exists $u \in C$ such that $f - ug = X^{h+1}f_1$ where $f_1 \in C$ and, by arguing by reductio ad absurdum deduce the existence of a process of "Euclidean division" in $C$.) If $s(f) = 0$, $f$ is invertible in $C$.

Show that $A$ is not a Dedekind domain.

(b) Deduce from (a) that, if $B$ is a Krull domain, $A = B[[X]]$ is also a Krull domain. (If $K$ is the field of fractions of $B$, note that $A$ is the intersection of $K[[X]]$ and a family (C_i) of principal ideal domains with the same field of fractions as $A$ and that every element of $A$ is invertible in almost all the $C_i$.)