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rational number

Definition

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

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 $d(x,y)≔\mid x-y\mid$; the completion is the field of real numbers.
3. Fixing a prime number $p$, the $p$-adic topology is defined by the ultrametric $d(x,y)≔1/n$ 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.)

Related entries

• natural number
• integer
• rational number
• algebraic number
• Gaussian number
• irrational number
• real number
• p-adic number