Editing Local field (section)

===Structure of the unit group===

The multiplicative group of non-zero elements of a non-Archimedean local field ''F'' is isomorphic to
:<math>F^\times\cong(\varpi)\times\mu_{q-1}\times U^{(1)}</math>
where ''q'' is the order of the residue field, and μ<sub>''q''−1</sub> is the group of (''q''−1)st roots of unity (in ''F''). Its structure as an abelian group depends on its [[characteristic (algebra)|characteristic]]:
*If ''F'' has positive characteristic ''p'', then
::<math>F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/{(q-1)}\oplus\mathbf{Z}_p^\mathbf{N}</math>
:where '''N''' denotes the [[natural number]]s;
*If ''F'' has characteristic zero (i.e. it is a finite extension of '''Q'''<sub>''p''</sub> of degree ''d''), then
::<math>F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/(q-1)\oplus\mathbf{Z}/p^a\oplus\mathbf{Z}_p^d</math>
:where ''a''&nbsp;≥&nbsp;0 is defined so that the group of ''p''-power roots of unity in ''F'' is <math>\mu_{p^a}</math>.{{sfn|Neukirch|1999|loc=Theorem II.5.7}}