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rational number

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)|xy|d(x,y) \coloneqq {|x - y|}; the completion is the field of real numbers. 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.)

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