# Rational numbers

## Definition

The field of rational numbers, $ℚ$, is the field of fractions of the commutative ring of integers, $ℤ$, hence the field consisting of formal fractions of integers.

## Properties

There are several interesting topologies on $ℚ$ that make $ℚ$ into a topological group under addition, allowing us to define interesting fields by taking the completion with respect to this topology:

1. The discrete topology is the most obvious, which is already complete.
2. The absolute-value topology is defined by the metric $d\left(x,y\right)≔\mid x-y\mid$; the completion is the field of real numbers.
3. Fixing a prime number $p$, the $p$-adic topology is defined by the ultrametric $d\left(x,y\right)≔1/n$ where $n$ is the highest exponent on $p$ in the prime factorization? of $\mid x-y\mid$; the completion is the field of $p$-adic numbers.

According to Ostrowski's theorem this are the only possibilities.

Interestingly, (2) cannot be interpreted as a localic group, although the completion $ℝ$ can. (Probably the same holds for (3); I need to check.)

Revised on November 7, 2013 01:57:26 by Urs Schreiber (188.200.54.65)