Local field

Remarks

A local field is Archimedean when it is $\R$ or $\C$, and non-Archimedean otherwise. Many authors use "local field" only for the non-Archimedean case. A non-Archimedean field is a local field exactly when it is complete for a discrete valuation and its residue field is finite. The non-Archimedean local fields of characteristic $0$ are the finite extensions of the fields $\Q_p$. Those of characteristic $p$ are the fields $\F_q((t))$ with $q$ a power of $p$. Example: $\Q_p$ is a local field, because the ball $\Z_p$ is a compact neighbourhood of $0$. Non-example: $\Q$ with $\abs{\cdot}_p$ is not a local field, because it is not complete, and a field with such an absolute value is locally compact only when complete. Non-example: $\C((t))$ is complete for a discrete valuation, but its residue field $\C$ is infinite, so it is not locally compact.