Field (mathematics)
Absolute Galois group

For fields that are not algebraically closed (or not separably closed), the absolute Galois group Gal(F) is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs all finite separable extensions of F. By elementary means, the group Gal(F) can be shown to be the Prüfer group, the profinite completion of Z. This statement subsumes the fact that the only algebraic extensions of Gal(F) are the fields Gal(F) for n > 0, and that the Galois groups of these finite extensions are given by

A description in terms of generators and relations is also known for the Galois groups of p -adic number fields (finite extensions of Q).

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. For example, the Brauer group, which is classically defined as the group of central simple F -algebras, can be reinterpreted as a Galois cohomology group, namely
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