Research papers in modern number theory often consider a more general notion, requiring only that the residue field be perfect of positive characteristic, not necessarily finite.
For a non-Archimedean local field F (with absolute value denoted by |·|), the following objects are important: Every non-zero element a of F can be written as a = ϖnu with u a unit, and n a unique integer.
The nth higher unit group of a non-Archimedean local field F is for n ≥ 1.
The multiplicative group of non-zero elements of a non-Archimedean local field F is isomorphic to where q is the order of the residue field, and μq−1 is the group of (q−1)st roots of unity (in F).
A non-Archimedean local field can be viewed as the field of fractions of the completion of the local ring of a one-dimensional arithmetic scheme of rank 1 at its non-singular point.
From the geometric point of view, n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n-dimensional arithmetic scheme.