Completion of a field at a place

Definition

Let $F$ be a field with an absolute value $|\cdot|_v$ representing a place $v$ . The completion of $F$ at $v$ is a complete valued field $F_v$ together with an isometric embedding

F\longrightarrow F_v

having dense image. It is unique up to a unique isometric isomorphism fixing $F$ .

Construction

The field $F_v$ is obtained by taking equivalence classes of Cauchy sequences for the metric $d_v(x,y)=|x-y|_v$ . Addition, multiplication, and the absolute value extend continuously from $F$ .

Global-field cases

If $F$ is a global field, every $F_v$ is a local field. For a number field, an archimedean completion is $\mathbb R$ or $\mathbb C$ , while a nonarchimedean completion is a finite extension of $\mathbb Q_p$ . For a global function field, it is a finite extension of a Laurent-series field over a finite field.

References

Jürgen Neukirch, Algebraic Number Theory, Springer, 1999, Chapter II, §§4--5.
Jean-Pierre Serre, Local Fields, Springer, 1979, Chapter I.