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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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  • Last edited by Andrew Sutherland on 2020-10-09 17:53:28
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