A place $v$ of a field $k$ is an equivalence class of non-trivial absolute values on $K$. As with absolute values, places may be classified as archimedean or nonarchimedean, since these properties are preserved under equivalence.
Each place induces a distance metric that gives $K$ a metric topology. The completion $K_v$ of $K$ at $v$ is the completion of this metric space, which is also a topological field.
When $K$ is a number field each nonarchimedean place arises from the valuation associated to each prime ideal in the ring of integers of $K$, while archimedean places arise from embeddings of $K$ into the complex numbers: each real embedding determines a real place, and each conjugate pair of complex embeddings determines a complex place. The archimedean places of a number field are also called infinite places.
- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-09 17:53:28
- 2020-10-09 17:53:28 by Andrew Sutherland (Reviewed)
- 2020-10-09 16:54:04 by Andrew Sutherland
- 2020-10-09 16:46:14 by Andrew Sutherland
- 2020-10-09 16:42:02 by Andrew Sutherland