Exercise 25, § 3

Let $k$ be a field of characteristic 2, $B$ the ring of formal power series $k[[X, Y]]$, $E$ its field of fractions and $A$ the $k$-derivation of $E$ defined by $\Delta(X) = Y^{2j}, \Delta(Y) = X^{2i}$ ($i, j$ integers $\geqslant 0$); the subring $A$ of $B$ consisting of the $x \in B$ such that $Ax = 0$ is the ring of formal power series in $k[[X, Y, Z]]$ where $X^2$ is substituted for $X$, $Y^2$ for $Y$ and $X^{2i+1} + Y^{2j+1}$ for $Z$.

(a) Show that the group $C(A)$ contains a vector space over $k$ of dimension $N(i, j)$ equal to the number of ordered pairs of integers $(a, b)$ such that $0 \leq a < i, 0 \leq b < j$ and $(2j+1)a + (2i+1)b \geq 2ij$. (Use Exercise 23 (b) and Exercise 24 (a), note that the elements of $L$ are the formal power series $F \in B$ such that $AF = F^2$; attributing to $X$ the weight $2j+1$ and to $Y$ the weight $2i+1$, decompose $F$ as an infinite sum of isobaric polynomials; if $L_r$ is the subgroup of $L$ consisting of the $F \in L$ whose isobaric components are of weight $\geq q$, $L/L'$ is isomorphic to the direct sum of the groups $C_q/C_{q+1}$, where $C_q = L_q/(L' \cap L_r)$; calculate these groups for $q \geq 4ij$, using Exercise 24 (b).)

(b) Show that the ideal $AX^2$ of $A$ is prime; if $A' = A[X^{-2}]$, deduce that $C(A')$ and $C(A)$ are isomorphic (no. 4, Proposition 3). Show that $A'$ is a Dedekind domain (consider the ring $B' = B[X^{-2}]$, which is integral over $A'$ and a principal ideal domain, and use Chapter V, § 2, no. 4, Theorem 3). Deduce an example of a Dedekind domain whose ideal class group is infinite.