Exercise 9, § 2

Let $k_0$ be a field and $A'$ the polynomial ring $k_0[X_n]_{n \in \mathbf{N}}$ in an infinite sequence of indeterminates; it is an integrally closed domain whose field of fractions is $K' = k_0(X_n)_{n \in \mathbf{N}}$. Let $\sigma$ denote the $k_0$-automorphism of $A'$ such that

for all $n \geq 0$; $\sigma$ and the identity automorphism form a group $\mathscr{G}$ of automorphisms of $A'$. Let $A$ be the ring of invariants of $\mathscr{G}$ and let $K$ be its field of fractions; $K'$ is a separable extension of $K$ of degree 2 and $A'$ is the integral closure of $A$ in $K'$.

(a) Show that $A'$ is an $A$-module admitting an infinite basis. (If $k_0$ is of characteristic $\neq 2$, show that, if we write $Y_n = X_{2n} - \frac{1}{2}X_{2n+1}$, $A$ is the subring of $A'$ generated by the $X_{2n+1}$ and the products $Y_n Y_m$ for $n > 0, m \geq 0$. If $k_0$ is of characteristic 2, $A'$ is generated by the $X_n, +_1$ and the

for $n \geq 0$.)

(b) Let $m'$ be the maximal ideal of $A'$ generated by all the $X_n$ and let $m = A \cap m'$; show that $m' = mA'$, that $m'$ is the only ideal of $A'$ lying above $m$ and that the fields $A/m$ and $A'/m'$ are canonically isomorphic.

(c) Let $p'$ be the prime ideal of $A'$ generated by the $X_{2n+1}$ for $n \geq 0$ and let $p = p' \cap A$; let $k, k'$ be the fields of fractions of $A/p$ and $A'/p'$ respectively; $p'$ is the only prime ideal of $A'$ lying above $p$. If $k_0$ is not of characteristic 2, $k'$ is a separable extension of $k$ of degree 2; if $k_0$ is of degree 2, $k'$ is an infinite radicial extension of $k$.