Exercise 4, § 9

(a) Let K be a commutative field and $v$ a valuation on K such that K is Henselian with $v$ ($§ 8$, Exercise 6). Suppose further that K and the residue field $k$ of $v$ are of characteristic 0. Show that there exists a subfield $K_0$ of the ring $\mathbf{A}$ of $v$ such that the canonical mapping $\mathbf{A} \to k$, restricted to $K_0$, is an isomorphism of $K_0$ onto $k$. (Let H be a subfield of $\mathbf{A}$ such that the image of H under the canonical mapping is an isomorphism onto a subfield E of $k$; show that, if $E \neq k$, there exists $\alpha \notin H$ in $\mathbf{A}$ such that the subfield $H(\alpha)$ of K is contained in $\mathbf{A}$ and is canonically isomorphic to $E(\bar{\alpha})$, where $\bar{\alpha}$ is the class of $\alpha$ in $k$; distinguish two cases according to whether $\bar{\alpha}$ is algebraic or transcendental over E.)

(b) Suppose further that $v$ is a discrete valuation and that K is complete with respect to $v$; deduce from (a) that K is isomorphic to the field of formal power series $k((T))$.