Archimedean local field

Definition

An archimedean local field is a local field whose absolute value is archimedean. Every such field is isomorphic as a topological field to $\mathbb R$ or $\mathbb C$ .

As a completion

For a number field $F$ , a real embedding $F\hookrightarrow\mathbb R$ gives a real place and completion $F_v\cong\mathbb R$ . A conjugate pair of nonreal embeddings $F\hookrightarrow\mathbb C$ gives a complex place and completion $F_v\cong\mathbb C$ .

Global function fields have no archimedean places.

Contrast with the nonarchimedean case

Archimedean local fields are connected Lie groups under addition. By contrast, nonarchimedean local fields are totally disconnected and have compact open valuation rings. This topological difference leads to different test functions and representation theory at the two kinds of places.

References

André Weil, Basic Number Theory, third edition, Springer, 1974, Chapter I.
Jürgen Neukirch, Algebraic Number Theory, Springer, 1999, Chapter II.