Algebraic Number Field - Galois Groups and Galois Cohomology

Galois Groups and Galois Cohomology

Generally in abstract algebra, field extensions F / E can be studied by examining the Galois group Gal(F / E), consisting of field automorphisms of F leaving E elementwise fixed. As an example, the Galois group Gal (Q(ζ_n) / Q) of the cyclotomic field extension of degree n (see above) is given by (Z/nZ)×, the group of invertible elements in Z/nZ. This is the first stepstone into Iwasawa theory.

In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extension F / F of the algebraic closure, leading to the absolute Galois group G := Gal(F / F) or just Gal(F), and to the extension F / Q. The fundamental theorem of Galois theory links fields in between F and its algebraic closure and closed subgroups of Gal (F). For example, the abelianization (the biggest abelian quotient) Gab of G corresponds to a field referred to as the maximal abelian extension Fab (called so since any further extension is not abelian, i.e., does not have an abelian Galois group). By the Kronecker–Weber theorem, the maximal abelian extension of Q is the extension generated by all roots of unity. For more general number fields, class field theory, specifically the Artin reciprocity law gives an answer by describing Gab in terms of the idele class group. Also notable is the Hilbert class field, the maximal abelian unramified field extension of F. It can be shown to be finite over F, its Galois group over F is isomorphic to the class group of F, in particular its degree equals the class number h of F (see above).

In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal(F), also known as Galois cohomology, which in the first place measures the failure of exactness of taking Gal(F)-invariants, but offers deeper insights (and questions) as well. For example, the Galois group G of a field extension L / F acts on L×, the nonzero elements of L. This Galois module plays a significant role in many arithmetic dualities, such as Poitou-Tate duality. The Brauer group of F, originally conceived to classify division algebras over F, can be recast as a cohomology group, namely H2(Gal (F), F×).