How to prove that $k(x)$ is not complete in the $x$-adic metric

Question

It is not hard to find proofs showing that $\mathbb{Q}$ is not complete with respect to the metric induced by the valuation $|\;\;|_p$. For example, it is enough to recall that every complete metric space is Baire and to show that $(\mathbb{Q},|\;\;|_p)$ is not Baire. Another option is to show that $\mathbb{Q}_p$ is uncountable, so $(\mathbb{Q},|\;\;|_p)$ is different than its completion. Both arguments use the fact that $\mathbb{Q}$ is countable.

But when I try to justify that $k(x)$ is not complete in the metric induced by the $x$-adic valuation $|\;\;|_x$, I don't find ideas or references addressing the proof. How can I proof the incompleteness of $(k(x), |\;\;|_x)$ without invoking its completion field $k((x))$? Any reference?

Answer 1

After trying and researching several days, I couldn't prove the result without invoking $K((x))$. So here is what I can do.

Definition: Let $K$ be a field. A sequence $(r_n)_n$ of elements of $K$ is linearly recurrent if there exist constants $c_1,c_2,\dots,c_k\in K$ such that \begin{equation}\label{linear recurrence} r_{n+k}=c_1r_{n}+c_2r_{n+1}+\cdots+c_kr_{n+k-1} \end{equation} for all $n\in\mathbb{N}$.

Theorem: Let $f(x)=\sum_{i=0}^\infty r_ix^i\in K[[x]]$, where $K$ is any field. For $m,s\geq0$, let $A_{s,m}$ be the matrix $\{r_{s+i+j}\}_{0\leq i,j\leq m}$:

\begin{pmatrix} r_s & r_{s+1} & r_{s+2} & \cdots & r_{s+m} \\ r_{s+1} & r_{s+2} & r_{s+3} & \cdots & r_{s+m+1} \\ r_{s+2} & r_{s+3} & r_{s+4} & \cdots & r_{s+m+2} \\ \vdots & \vdots & \vdots & & \vdots \\ r_{s+m} & r_{s+m+1} & r_{s+m+2} & \cdots & r_{s+2m} \end{pmatrix}

and let $N_{s,m}:=\det(A_{s,m})$. Then $f(x)\in K(x)$ if and only if there exist integers $m\geq0$ and $S$ such that $N_{s,m}=0$ whenever $s\geq S$, or equivalently, if the sequence $(r_n)_n$ is eventually linearly recurrent.

Proof: See V.5. Lemma 5 of Koblitz, N., "p-adic Numbers, p-adic Analysis, and Zeta-Functions." Vol. 58. Springer Science and Business Media, 2012. Also see III.3.1.N of Ribenboim, P., "The theory of classical valuations." Springer Science and Business Media, 1999.

Now we are in position to prove the incompleteness of $(K(x),|\;\;|_x)$. Let $f(x)=\sum_{i=0}^\infty x^{i^2}\in K((x))$. Notice that $f(x)$ is the limit of a sequence of rational functions, say $a_n(x)=\sum_{i=0}^n x^{i^2}\in K[x]$. Thus $(a_n(x))_n$ is a Cauchy sequence in $K(x)$. We will show that $(a_n(x))_n$ is not convergent in $K(x)$ by showing that $f(x)\not\in K(x)$.

Given $m$ and $S$ non-negative integers, choose $n\in\mathbb{N}$ such that $S<(n-1)^2<n^2-m<n^2<n^2+m<(n+1)^2$. By putting $\ell:=n^2-m$ we have $$S<(n-1)^2<\ell<\ell+m=n^2<\ell+2m<(n+1)^2.$$ Since all the members of an anti-diagonal of $A_{\ell,m}$ are equal, it follows that, $$A_{\ell,m}= \begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 & 1 & 0 \\ 0 & 0 & 0 & \cdots & 1 & 0 & 0 \\ \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\ 0 & 0 & 1 & \cdots & 0 & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 0 & 0 & \cdots & 0 & 0 & 0 \end{pmatrix} $$

and $N_{\ell,m}\neq0$ with $\ell>S$. Then by Theorem, $f(x)\not\in K(x).$