Editing Local field (section)

===<span id="higherunit"></span><span id="principalunit"></span>Higher unit groups===

The '''''n''<sup>th</sup> higher unit group''' of a non-Archimedean local field ''F'' is
:<math>U^{(n)}=1+\mathfrak{m}^n=\left\{u\in\mathcal{O}^\times:u\equiv1\, (\mathrm{mod}\,\mathfrak{m}^n)\right\}</math>
for ''n''&nbsp;≥&nbsp;1. The group ''U''<sup>(1)</sup> is called the '''group of principal units''', and any element of it is called a '''principal unit'''. The full unit group <math>\mathcal{O}^\times</math> is denoted ''U''<sup>(0)</sup>.

The higher unit groups form a decreasing [[filtration (mathematics)|filtration]] of the unit group
:<math>\mathcal{O}^\times\supseteq U^{(1)}\supseteq U^{(2)}\supseteq\cdots</math>
whose [[quotient group|quotients]] are given by
:<math>\mathcal{O}^\times/U^{(n)}\cong\left(\mathcal{O}/\mathfrak{m}^n\right)^\times\text{ and }\,U^{(n)}/U^{(n+1)}\approx\mathcal{O}/\mathfrak{m}</math>
for ''n''&nbsp;≥&nbsp;1.{{sfn|Neukirch|1999|p=122}} (Here "<math>\approx</math>" means a non-canonical isomorphism.)