Local field

Summary

A local field is a field with a nontrivial absolute value for which it is locally compact. The Archimedean local fields are $\R$ and $\C$. The others are the finite extensions of the fields $\Q_p$ of p-adic numbers and the fields $\F_q((t))$ of Laurent series over finite fields. They are the completions of global fields. Number theory studies a global field one place at a time through its local fields.