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, $\Delta$ the $k$-derivation of $E$ defined by $\Delta(X) = Y^{2j}$, $\Delta(Y) = X^{2i}$ ($i, j$ integers ≥ 0); the subring $A$ of $B$ formed by the $x \in B$ such that $\Delta x = 0$ is the ring of formal power series of $k[[X, Y, Z]]$ where one substitutes $X^2$ for $X$, $Y^2$ for $Y$ and $X^{2i+1} + Y^{2j+1}$ for $Z$.

(a) Now prove that the group $C(A)$ contains a k-vector space of dimension $N(i, j)$, equal to the number of pairs of integers $(a, b)$ such that $0 \le a < i$, $0 \le b < j$ and $(2j + 1)a + (2i + 1)b \ge 2ij$. (Using Ex. 23 b) and Ex. 24 a), note that the elements of $L$ are the formal power series $F \in B$ such that $\Delta F = F^2$; by assigning to $X$ the weight $2j + 1$ and to $Y$ the weight $2i + 1$, decompose $F$ into an infinite sum of isobaric polynomials; if $L_q$ is the subgroup of $L$ formed by the $F \in L$ whose isobaric components are of weight ≥ $q$, $L/L’$ is isomorphic to the direct sum of the groups $C_q/C_{q+1}$, where $C_q = L_q/(L’ \cap L_q)$; calculate these groups for $q \ge 4ij$, using Ex. 24 b)).

(b) Now prove 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, Prop. 3). Now prove that $A’$ is a Dedekind ring (consider the ring $B’ = B[X^{-2}]$, which is integral over $A’$ and is principal, and use Chapter V, § 2, No. 4, Theorem 3). Deduce an example of a Dedekind ring whose group of ideal classes is infinite.