Global and Local Fields; Completions

A global field (in this letter) is a number field, i.e. a finite extension $F/\mathbb{Q}$.

A place $v$ of $F$ is an equivalence class of absolute values on $F$; it is archimedean if it comes from an embedding $F\hookrightarrow \mathbb{R}$ or $\mathbb{C}$, and nonarchimedean otherwise.

The completion $F_v$ is the completion of $F$ with respect to $|\cdot|_v$.

A local field is a nondiscrete locally compact topological field; examples: $\mathbb{R}$, $\mathbb{C}$, and finite extensions of $\mathbb{Q}_p$.