transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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.
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:
The discrete topology is the most obvious, which is already complete.
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.))
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.)
rational number
a finite field extension of $\mathbb{Q}$ is called a number field