Exercise 5, § 5

(a) Let K be a field, $v$ a valuation on K and $\Gamma$ the order group of $v$. For K to be linearly compact (Chapter III, § 2, Exercise 15) with the topology $\mathscr{T}_v$, it is necessary and sufficient that, for every well-ordered subset B of $\Gamma$ and every family $(a_\beta)_{\beta \in B}$ of elements of K such that, for $A < \mu < v$,

there exists an element $a \in K$ such that $v(a - a_\lambda) = v(a_\lambda - a)$ for every ordered pair of indices $A, \mu$ such that $\mu > A$. (Use Exercise 4 of Set Theory,

Chapter 111, § 2). If this is so, the ring A of the valuation v is also linearly compact with the discrete topology.

(b) Show that the field S(\Gamma, k) = K defined in § 3, Exercise 2 is linearly compact with $\mathscr{T}_v$. Take \Gamma to be the totally ordered group \mathbf{Q} of rational numbers and consider the subring K_0 of K consisting of the $x = (x_\alpha)$ such that the set of $a \in \mathbf{Q}$ for which $x, \# 0$ is finite or is the set of points of a strictly increasing sequence tending to $+\infty$. Show that K_0 is a field which is complete with respect to the valuation $v_0$ induced by the canonical valuation v on K and that $v_0$ has the same value group and residue field as v but that K_0 is not linearly compact (cf. § 10, Exercise 2).