Exercise 20, § 1

Let $K_0$ be a perfect field of characteristic $p > 0$, $K$ the field of rational functions $K_0(X_n)_{n \in \mathbf{N}}$ and $B$ the ring of formal power series $K[[Z]]$, where $Z$ is an indeterminate; if $m$ is the maximal ideal of $B$, $B$ is Hausdorff and complete with the $m$-adic topology and is a discrete valuation ring. Let $L = K((Z))$ be the field of fractions of $B$ and $E$, the subfield of $L$ generated by $L^p$ and the elements $Z$ and $X_n$ (for $n \in \mathbf{N}$).

(a) Show that the element $c = \sum_{n=0}^\infty X_n Z^n$ of $L$ does not belong to $E$.

(b) Let $E$ be a maximal element of the set of subfields of $L$ containing $E$, and not containing $c$; we write $A = B \cap E$. Show that $A$ is a discrete valuation ring (and therefore Noetherian) whose field of fractions is $E$; if $m'$ is the maximal ideal of $A$, then ${m'}^k = m^k \cap A$ and $A$ is dense in $B$ with the $m$-adic topology; $L = K(c)$ is an extension of $K$ of degree $p$ and $B$ is the integral closure of $A$ in $L$ but $B$ is not a finitely generated $A$-module (use Chapter 111, § 3, Exercise 9 (b)). \*