Exercise 8, § 5

(a) Let $K$ be a field, $v$ a valuation on $K$ of height 1 and $A$ its ring. Let $H$ be a compact subset of $K$ (with the topology $\mathscr{T}_v$) and $a \neq 0$ a point of $K$. Show that there exists a polynomial $f \in K[X]$ without constant term and such that $f(a) = 1, f(H) \subset A$. (Prove that $f$ may be taken to be a polynomial of the form

where the $c_i$ are suitably chosen elements of $H$ such that $v(c_i) < v(a)$ and the $n(i)$ are sufficiently large integers $> 0$.)

(b) Let $X$ be a totally disconnected compact space; let the ring $C(X; K)$ of continuous mappings from $X$ to $K$ be given the uniform convergence topology. Let $B$ be a subring of $C(X; K)$ containing the constants and separating the points of $X$; show that $B$ is dense in $C(X; K)$ (Use (a) and General Topology, Chapter X, § 4, Exercise 21 (b).)