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**.

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- Review status: reviewed
- Last edited by John Jones on 2021-07-06 12:16:11

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- 2021-07-06 12:16:11 by John Jones (Reviewed)
- 2021-06-06 10:47:20 by John Jones (Reviewed)
- 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

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