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A local field is a field $K$ with a non-trivial absolute value $|\ |$ that is locally compact in the topology induced by the distance metric $d(x,y):=|x-y|$.

An archimedean local field is a local field whose absolute value is archimedean; such a field is isomorphic to $\R$ or $\C$.

A nonarchimedean local field is a local field whose absolute value is nonarchimedean; such fields are isomorphic to finite extensions of $\Q_p$ when $K$ has characteristic zero, and to a finite extension of $\F_p((t))$ when $K$ characteristic $p$. In both cases $p$ is the characteristic of the residue field.

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• Last edited by Andrew Sutherland on 2020-10-24 17:22:32
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