Notions of convergence over extensions of finite fields

Question

Let $\displaystyle Q_p[x] = \left\{\frac{p(x)}{q(x)} \mid \, p(x),q(x) \in \mathbb{F}_p[x], \, q(x) \neq 0 \right\}$ denote the field of fractions extending $\mathbb{F}_p[x]$. If we consider the sequence $\displaystyle \left\{\sum_{k=0}^n x^i\right\}$, that, in $\mathbb{R}$, would converge to $\displaystyle \frac{1}{1-x}$ with the usual metric $| \cdot|$ kept in mind, then, say, over elements in $\mathbb{F}_2$, what metric can we construct such that we make sense of the "intuitive limit" (or what should feel like) $$\sum_{k=0}^\infty x^k = \frac{1}{1+x} \, ?$$

If this has been done so before, references would very much be appreciated. In regards to what I have done, I have entertained the use of the Hamming metric, but its application only makes sense in $\mathbb{F}_p[x]$ and I'm forced to construct an extension of the Hamming metric for $Q_p[x]$, which I have not done so successfully (or at least acceptably so). Any suggestions?

Answer 1

A few things: 1) The usual notation for the field of rational functions over a field is to use parentheses, so the field you're looking for is denoted $\mathbb{F}_p(x)$.

2) The field of rational functions isn't the extension of $\mathbb{F}_p[x]$ you probably want, as it includes $x^{-1}$, which can't be limited to in your sense; the extension you want probably should be $\mathbb{F}_p[[x]]$, the ring of formal power series $\sum_i a_i x^i$ over $\mathbb{F}_p$. This ring has a valuation, allowing the concept of limits to make sense. The term you may want would be "valuation ring". And in this ring, $(1-x) \sum x^i = 1$.

3) There is a field $\mathbb{F}_p((x))$ extending both; it is the field of fractions of $\mathbb{F}_p[[x]]$. One way to think of it is as the formal power series "starting" at some minimum power of $x$. Every rational function can be expressed as an element of this field.