Exercise 2, § 10

Let $K$ be a field, $v$ a valuation on $K$, $L$ an extension of $K$ and $v'$ a valuation on $L$ extending $v$; $L$ is called an immediate extension (with respect to $u'$) if $e(v'/v) = f(v'/v) = 1$. The completion $\hat{K}$ of $K$ is an immediate extension of $K$.

(a) For $L$ to be an immediate extension of $K$, it is necessary and sufficient that, for all $x \in L - K$, there exist $y \in K$ such that $v'(x - y) > v'(x)$.

(b) For every field $K$, show that there exists a maximal immediate extension $L$ of $K$, that is, not admitting an immediate extension distinct from itself (use § 3, Exercise 3 (b)).

(c) Show that, if $K$ is linearly compact with the topology defined by $v$, it admits no immediate extension other than itself (use (a), noting that, in the notation of (a), the set of $v'(x - y)$ for $y \in K$ has no greatest element in the value group of $v'$).

(d) Suppose that $K$ is not linearly compact; let $B$ be a well-ordered subset of the order group of $v$ and $(a_\beta)_{\beta \in B}$ a family of elements of $K$ such that, for $\lambda < \mu < v$,

but that there exists no $x \in K$ such that $v(x - a_\lambda) = v(a_\lambda - a_\gamma)$ for every ordered pair $(\lambda, \mu)$ such that $\lambda < \mu$. For every extension $E$ of $K$ and every extension $w$ of $v$ to $E$, let $D_E(\lambda)$ denote the set of $z \in E$ such that $v(z - a_\lambda) \geq \gamma_\lambda$, where $\gamma_\lambda$ is the common value of the $v(a_\lambda - a_\gamma)$ for $\lambda < \mu$; the hypothesis is therefore that $\bigcap_{\beta \in B} D_K(\beta) = \varnothing$ (§ 5, Exercise 5 (a)). Let $\Omega$ be an algebraic closure of $K$ and let $v_0$ be an extension of $v$ to $\Omega$; show that, if $P$ is a polynomial in $K[X]$, for there to exist $A \in B$ such that $v(P(a_\mu)) = v(P(a_\lambda))$ for all $\lambda \leq \mu$, it is necessary and sufficient that no zero of $P$ in $\Omega$ belong to $\bigcap_{\beta \in B} D_\Omega(\beta)$. If $P$ has this property and $E$ and $w$ have the same meaning as above, show that, for all $z \in \bigcap_{\beta \in B} D_E(\beta)$, $w(P(z) - P(a_\mu)) > w(P(z)) = v(P(a_\mu))$ so long as $\mu$ is large enough (decompose $P(X)$ into factors in $\Omega[X]$). Deduce that one of the following situations holds:

(1) Either $\bigcap_{\beta \in B} D_\Omega(\beta) \neq \varnothing$ and, if $\theta$ is an element of this intersection whose degree over $K$ is the least possible, $K(\theta)$ is an immediate extension of $K$ distinct from $K$.

such that $\bigcap_{\mu} D_\alpha(\beta) = \varnothing$; then there is a valuation $v'$ on $K(X)$ extending $v$ immediate extension of $K$ (use the criterion in (a)).

(e) Deduce from (c) and (d) that for $K$ to be linearly compact, it is necessary and sufficient that $K$ have no immediate extension other than itself.