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In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper In Mathematics, and in particular in Algebraic number theory, a Galois module is a module for a Galois group G. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a Galois group is a group associated with a certain type of Field extension. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. In Mathematics, more specifically in Abstract algebra, field extensions are the main object of study in field theory. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, and in particular in Algebraic number theory, a Galois module is a module for a Galois group G. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. In Homological algebra, an exact functor is a Functor, from some category to another which preserves Exact sequences Exact functors are very

The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of idele class groups in algebraic number theory was one way to formulate class field theory, at the time in the process of ridding itself of connections to L-functions. In Mathematics, an adelic algebraic group is a Topological group defined by an Algebraic group G over a Number field K In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers In Mathematics, class field theory is a major branch of Algebraic number theory. The theory of L -functions has become a very substantial and still largely Conjectural, part of contemporary Number theory. Galois cohomology makes no assumption that Galois groups are abelian groups, so that this was a non-abelian theory. It was formulated abstractly as a theory of class formations. In mathematics a class formation is a structure used to organize the various Galois groups and modules that appear in Class field theory. Two developments of the 1960s turned the position around. Firstly, Galois cohomology appeared as the foundational layer of étale cohomology theory (roughly speaking, the theory as it applies to zero-dimensional schemes). In Mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological Secondly, non-abelian class field theory was launched as part of the Langlands philosophy, which meant that L-functions were back, with a vengeance. In Mathematics, non-abelian class field theory is a catchphrase meaning the extension of the results of Class field theory, the relatively complete and classical The Langlands program is a web of far-reaching and influential Conjectures that connect Number theory and the representation theory of certain groups

The earliest results identifiable as Galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. In Mathematics, the arithmetic of abelian varieties is the study of the Number theory of an Abelian variety, or family of those The normal basis theorem implies that the first cohomology group of the additive group of L will vanish; this is a result on general field extensions, but was known in some form to Dedekind. In Mathematics, a normal basis in field theory is a special kind of basis for Galois extensions of finite degree characterised as forming a single In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important The corresponding result for the multiplicative group is known as Hilbert's Theorem 90, and was known before 1900. In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Number theory, Hilbert's Theorem 90 (or Satz 90) refers to an important result on Cyclic extensions of Number fields (or to one of its generalizations Kummer theory was another such early part of the theory, giving a description of the connecting homomorphism coming from the m-th power map. In Mathematics, Kummer theory provides a description of certain types of Field extensions involving the adjunction of n th roots of elements of

In fact for a while the multiplicative case of a 1-cocycle for groups that are not necessarily cyclic was formulated as the solubility of Noether's equations, named for Emmy Noether; they appear under this name in Emil Artin's treatment of Galois theory, and may have been folklore in the 1920s. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician The case of 2-cocycles for the multiplicative group is that of the Brauer group, and the implications seem to have been well known to algebraists of the 1930s. In Mathematics, the Brauer group arose out of an attempt to classify Division algebras over a given field K.

In another direction, that of torsors, these were already implicit in the infinite descent arguments of Fermat for elliptic curves. In Mathematics, a principal homogeneous space, or torsor, for a group G is a set X on which G acts freely and In Mathematics, a proof by infinite descent is a particular kind of proof by Mathematical induction. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O Numerous direct calculations were done, and the proof of the Mordell-Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H1 group. In Mathematics, the Mordell–Weil theorem states that for an Abelian variety A over a Number field K, the group A ( The 'twisted' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups (such as quadratic forms, simple algebras, Severi-Brauer varieties), in the 1930s, before the general theory arrived. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, specifically in Ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the set { In Mathematics, a Severi-Brauer variety over a field K is an Algebraic variety V which becomes Isomorphic to Projective

The needs of number theory were in particular expressed by the requirement to have control of a local-global principle for Galois cohomology. In Mathematics, Helmut Hasse 's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation This was formulated by means of results in class field theory, such as Hasse's norm theorem. In Number theory, the Hasse norm theorem states that if L/K is a Cyclic extension of Number fields then if a nonzero element of K is a local norm everywhere In the case of elliptic curves it led to the key definition of the Tate-Shafarevich group in the Selmer group, which is the obstruction to the success of a local-global principle. In Mathematics, particularly in Arithmetic geometry, the Weil-Châtelet group of an Abelian variety A defined over a field K In Mathematics, particularly in Arithmetic geometry, the Weil-Châtelet group of an Abelian variety A defined over a field K Despite its great importance, for example in the Birch and Swinnerton-Dyer conjecture, it proved very difficult to get any control of it, until results of Karl Rubin gave a way to show in some cases it was finite (a result generally believed, since its conjectural order was predicted by an L-function formula). In Mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the Abelian group of points over a Number field of an Elliptic

The other major development of the theory, also involving John Tate was the Tate-Poitou duality result. John Torrence Tate Jr born March 13, 1925 in Minneapolis Minnesota, is an American Mathematician, distinguished for many fundamental

Technically speaking, G may be a profinite group, in which case the definitions need to be adjusted to allow only continuous cochains. In Mathematics, profinite groups are Topological groups that are in a certain sense assembled from Finite groups they share many properties with their finite

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