There are several interesting topologies on that make 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 ; the completion is the field of real numbers. 3. Fixing a prime number , the -adic topology is defined by the ultrametric where is the highest exponent on in the prime factorization? of ; the completion is the field of -adic numbers.
According to Ostrowski's theorem this are the only possibilities.
Interestingly, (2) cannot be interpreted as a localic group, although the completion can. (Probably the same holds for (3); I need to check.)