Metric completion of field of fractions The integers have as a field of fractions the rational numbers which have a metric completion as the real numbers. The reals can be represented by infinite decimal expansions which can be approximated by finite decimal expansions and some real numbers have algorithms which can generate the digits but some reals are uncomputable and some are undefinable.
If instead you start with a polynomial ring, $\Bbb{Q}[X]$ say, and form the field of rational functions $\Bbb{Q}(X)$, what metric completions of this fraction field exist and how can they be represented?
For background see profile.
 A: Metric spaces have but one completion. It is a unique metric space in which the original space is dense and it is complete.
The real numbers are the completion of the rationals when those are metricized by the absolute value; but there are other ways to give a metric to the rational numbers which yields different completions, examples are the $p$-adic numbers.
In order to discuss a metric completion of $\Bbb Q(X)$, or rather the metric completion you first need to specify a metric. Do note that $\Bbb Q(X)$ is a transcendental extension of $\Bbb Q$ and as such can be embedded into the real numbers so there are many metrics which end up being completed as the real numbers. 
A: Note that the metric completion will depend on what metric you put on the field. For instance, if we imbue the rationals $\bf Q$ with the p-adic metric, we obtain ${\bf Q}_p$, the $p$-adic numbers, upon our completion procedure. See http://en.wikipedia.org/wiki/P-adic_number.
Similarly, the field of fractions of a polynomial ring ${\bf F}[T]$ over a field $\bf F$ will be the "function field" denoted ${\bf F}(T)$ consisting of rational functions in $T$. Note that every rational function in $T$ can be expressed formally as an infinite Laurent expansion in $T$ using the geometric sum formula,
$$(a\ne0)\qquad \frac{1}{a-T}=\frac{1}{a}\frac{1}{1-a^{-1}T}=\frac{1}{a}\left(1+a^{-1}T+a^{-2}T^2+\cdots\right).$$
When we have the right metric defined on ${\bf F}(T)$, the above infinite series can be made sense of since it converges. Namely, define the value $|T^nf(T)|=e^{-n}$ when $f(0)\ne0$, and extend this to the rest of ${\bf F}(T)$ by multiplicativity (note the base $e$ was arbitrary; it can be normalized otherwise, and there are standard normalizations as $|\bf F|$ or similar when $\bf F$ is finite). This value induces a metric.
Under this metric, the completion of ${\bf F}[T]$ is ${\bf F}[[T]]$, the ring of infinite power series expansions in the variable $T$ (nonnegative powers only), and the completion of ${\bf F}(T)$ is ${\bf F}((T))$, the ring of formal Laurent expansions in $T$ (so, power series with finitely many negative powers allowed).
Here's how to see this for ${\bf F}[T]$. Suppose $f_m(T)$ is Cauchy with respect to $|\cdot|$. Then for every $n\ge0$, the coefficient $c_n(f_m)$ of $f_m(T)$ evenetually stabilizes (i.e. remains fixed for all $m$ beyond a certain point), since for every $n\ge0$ there is a $M$ for which $m_1,m_2>M\implies T^{n+1}\mid f_{m_1}(T)-f_{m_2}(T)$ or equivalently $f_{m_1}(T),f_{m_2}(T)$ share the same initial $n$ coefficients. Thus as we let $n\to\infty$ and view the polynomials $f_n(T)$ as finite power series, eventually every coefficient will be fixed and the power series consisting of all those stabilized coefficients is the resulting limit. Conversely, you can pick any power series out of ${\bf F}[[T]]$ and its partial sums will converge to it from ${\bf F}[T]$.
A similar argument goes for ${\bf F}(T)$ and ${\bf F}((T))$; we can use partial fraction decomposition to view the function field ${\bf F}((T))$ as a subring of ${\bf F}[[T]][T^{-1}]$ and do the same thing (extending the metric).
These are not the only metrics and thus not the only completions possible. Analogous to the number field setting, there will be $\pi$-adic absolute values on ${\bf F}[T]$ when $\pi(T)$ is an irreducible polynomial, every polynomial will factor into irreducibles $f(T)=\pi_0(T)^{s_0}\cdots\pi_r(T)^{s_r}$ and we then define $|f(T)|_\pi=e^{-(\deg\pi)s_0}$, where $\pi_0(T):=\pi(T)$ (again, this can be renormalized). This can be checked to be a non-archimedean absolute value and again we get a(n ultra)metric. Just like integers have base-$p$ expansions, we may write $f(T)=a_n(T)\pi(T)^n+\cdots+a_1(T)\pi(T)+a_0(T)$ with each $\deg a_i(T)<\deg\pi(T)$, using the division algorithm.
As a consequence, the completion of ${\bf F}[T]$ with respect to $|\cdot|_\pi$ can be represented as the ring of infinite power series in $\pi(T)$, and similarly the $\pi$-adic completion of ${\bf F}(T)$ allows negative powers.
Even the $\pi$-adic absolute values (and thus metrics) are not the only ones that can be defined; another standard one is $|f(T)|=e^{\deg f}$, which is kind of like the "$1/T$-adic" abs. val. If we let $\alpha$ be an arbitrary transcendental in $\bf C$, then ${\bf Q}(T)$ can be embedded as field in $\bf C$ and inherits the metric from the complex numbers on it, thus giving ${\bf Q}(T)$ and archimedean absolute value. We can, however, classify the absolute values ${\bf F}(T)$ that restrict trivially, analogous to Ostrowski's theorem.
