Editing Field (mathematics) (section)

=== Absolute Galois group ===
For fields that are not algebraically closed (or not separably closed), the [[absolute Galois group]] {{math|Gal(''F'')}} is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs ''all'' finite separable extensions of {{math|''F''}}. By elementary means, the group {{math|Gal('''F'''<sub>''q''</sub>)}} can be shown to be the [[Prüfer group]], the [[profinite completion]] of {{math|'''Z'''}}. This statement subsumes the fact that the only algebraic extensions of {{math|Gal('''F'''<sub>''q''</sub>)}} are the fields {{math|Gal('''F'''<sub>''q''<sup>''n''</sup></sub>)}} for {{math|''n'' > 0}}, and that the Galois groups of these finite extensions are given by
: {{math|1=Gal('''F'''<sub>''q''<sup>''n''</sup></sub> / '''F'''<sub>''q''</sub>) = '''Z'''/''n'''''Z'''}}.
A description in terms of generators and relations is also known for the Galois groups of {{math|''p''}}-adic number fields (finite extensions of {{math|'''Q'''<sub>''p''</sub>}}).<ref>{{harvp|Jannsen|Wingberg|1982}}</ref>

[[Galois representation|Representations of Galois groups]] and of related groups such as the [[Weil group]] are fundamental in many branches of arithmetic, such as the [[Langlands program]]. The cohomological study of such representations is done using [[Galois cohomology]].<ref>{{harvp|Serre|2002}}</ref> For example, the [[Brauer group]], which is classically defined as the group of [[central simple algebra|central simple {{math|''F''}}-algebras]], can be reinterpreted as a Galois cohomology group, namely
: {{math|1=Br(''F'') = H<sup>2</sup>(''F'', '''G'''<sub>m</sub>)}}.