Exercise 3, § 8

Let $k$ be the field of rational functions $\mathbf{F}_p(X_n)_{n \in \mathbf{N}}$ in an infinity of indeterminates over the prime field $\mathbf{F}$, and let $K = k(U, V)$ be the field of rational functions in two indeterminates over $k$.

(a) Show that the element $P(U) = \sum_{n=0}^\infty X_n^p U^{np}$ of the field of formal power series $k((U))$ is not algebraic over the field $k(U)$ of rational functions. The mapping $F(U, V) \mapsto F(U, P(U))$ of $k[U, V]$ to $k((U))$ can be extended to an isomorphism of $K$ onto a subfield of $k((U))$; the restriction to this subfield of the valuation on $k((U))$ equal to the order of the formal power series (§ 3, no. 3, Example 3 ) is a discrete valuation $v$ on $K$, whose residue field is $k$.

(b) Let $K'$ be the algebraic extension $K(V^{1/p})$ of $K$ so that $[K': K] = p$; if $v'$ is the unique valuation on $K'$ extending $v$, show that $e(v'/v) = f(v'/v) = 1$. The ring of the valuation $v'$ is therefore not a finitely generated module over the ring of the valuation $v$.