Definition
Local field
A nondiscrete locally compact topological field.
Definition
A local field is a nondiscrete Hausdorff topological field whose underlying space is locally compact.
Under this convention, the local fields are the archimedean local fields and , together with the nonarchimedean local fields.
Convention warning
Some authors reserve “local field” for the nonarchimedean cases. Statements in harmonic analysis and the Langlands program should therefore say “archimedean” or “nonarchimedean” whenever the distinction affects the objects under discussion.
Relation to global fields
The completion of a global field at a place is a local field. Conversely, every local field arises as such a completion, though not canonically from a unique global field.
References
- André Weil, Basic Number Theory, third edition, Springer, 1974, Chapter I.
- Jean-Pierre Serre, Local Fields, Springer, 1979, Chapter I.