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â„“-adic Representation
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
