Nonarchimedean local field

Definition

A nonarchimedean local field is a field $F$ complete with respect to a nontrivial discrete valuation whose residue field is finite.

Equivalently, it is a nondiscrete locally compact field whose topology is defined by a nonarchimedean absolute value.

Classification by characteristic

In characteristic $0$ , $F$ is a finite extension of $\mathbb Q_p$ for a unique prime $p$ , hence a $p$ -adic field. In positive characteristic, $F$ is isomorphic to a finite extension of $\mathbb F_q((t))$ .

Basic local data

The valuation determines a valuation ring $\mathcal O_F$ , its maximal ideal $\mathfrak p_F$ , a finite residue field $k_F=\mathcal O_F/\mathfrak p_F$ , and a uniformizer. The ring $\mathcal O_F$ is compact and open, so the additive and multiplicative groups of $F$ are locally profinite.

References

Jean-Pierre Serre, Local Fields, Springer, 1979, Chapter I.
André Weil, Basic Number Theory, third edition, Springer, 1974, Chapter I.