rational number

Rational numbers

Rational numbers


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.


Algebraic closure

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


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)|xy|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 pp, the pp-adic topology is defined by the ultrametric d(x,y)1/nd(x,y) \coloneqq 1/n where nn is the highest exponent on pp in the prime factorization of |xy|{|x - y|}; the completion is the field of pp-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 March 28, 2021 at 02:50:01. See the history of this page for a list of all contributions to it.