# Rational numbers

## Definition

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

## Properties

There are several interesting topologies on $\mathbb{Q}$ that make $\mathbb{Q}$ 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 $d(x,y) \coloneqq {|x - y|}$; the completion is the field of real numbers. 3. Fixing a prime number $p$, the $p$-adic topology is defined by the ultrametric $d(x,y) \coloneqq 1/n$ where $n$ is the highest exponent on $p$ in the prime factorization? of ${|x - y|}$; 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 $\mathbb{R}$ can. (Probably the same holds for (3); I need to check.)

Revised on November 6, 2014 19:51:31 by Urs Schreiber (81.194.35.225)