Rational numbers

Rational numbers

Definition

A rational number is a fraction of two integer numbers.

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 (“ratios”) of integers.

Properties

Algebraic closure

The algebraic closure $\overline{\mathbb{Q}}$ of the rational numbers is called the field of algebraic numbers. The absolute Galois group $Gal(\overline{\mathbb{Q}}\vert \mathbb{Q})$ has some curious properties, see there.

Topologies

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.

(This topology is totally disconnected (this exmpl.))

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.)

Last revised on September 25, 2018 at 11:34:06. See the history of this page for a list of all contributions to it.