|
Hasse 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, then it is a global norm. Here to be a global norm means to be an element ''k'' of K such that there is an element ''l'' of L with \mathbf_(l) = k; in other words ''k'' is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then ''k'' is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean. The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension (\sqrt,\sqrt)/ but is not a global norm. Serre and Tate showed that another counterexample is given by the field (\sqrt,\sqrt)/ where every rational square is a local nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Description Class field theory provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either definition, is always abelian. If a field ''K'' contains a primitive ''n''-th root of unity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local-global Principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an diophantine equation, integer solution to an equation by using the Chinese remainder theorem to piece together solutions modular arithmetic, modulo powers of each different prime number. This is handled by examining the equation in the Completion (ring theory), completions of the rational numbers: the real numbers and the p-adic number, ''p''-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers ''and'' in the ''p''-adic numbers for each prime ''p''. Intuition Given a polynomial equation with rational coefficients, if it has a rational solution, then this also yields a real solution and a ''p''-adic solution, as the rationals embed in the reals and ''p''-adics: a global solution yields local solutions at each prime. The Hasse principle a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Cohomology
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. A Galois group ''G'' associated with 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. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adelic Algebraic Group
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the definition of the appropriate topology is straightforward only in case ''G'' is a linear algebraic group. In the case of ''G'' being an abelian variety, it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms. In case ''G'' is a linear algebraic group, it is an affine algebraic variety in affine ''N''-space. The topology on the adelic algebraic group G(A) is taken to be the subspace topology in ''A''''N'', the Cartesian product of ''N'' copies of the adele ring. In this case, G(A) is a topological group. Histor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Cohomology Group
In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by , and are used in class field theory. Definition If ''G'' is a finite group and ''A'' a ''G''-module, then there is a natural map ''N'' from H_0(G,A) to H^0(G,A) taking a representative ''a'' to \sum_ ga (the sum over all ''G''-conjugates of ''a''). The Tate cohomology groups \hat H^n(G,A) are defined by *\hat H^n(G,A) = H^n(G,A) for n\ge 1, *\hat H^0(G,A)=\operatorname N= quotient of H^0(G,A) by norms of elements of ''A'', *\hat H^(G,A)=\ker N= quotient of norm 0 elements of ''A'' by principal elements of ''A'', *\hat H^(G,A) = H_(G,A) for n\le -2. Properties * If :: 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 :is a short exact sequence of ''G''-modules, then we get the usual long exact sequence of Tate cohomology groups: ::\cdots \longrightarrow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grunwald–Wang Theorem
In algebraic number theory, the Grunwald–Wang theorem is a local-global principle stating that—except in some precisely defined cases—an element ''x'' in a number field ''K'' is an ''n''th power in ''K'' if it is an ''n''th power in the completion K_ for all but finitely many primes \mathfrak of ''K''. For example, a rational number is a square of a rational number if it is a square of a number for almost all prime numbers ''p''. It was introduced by , but there was a mistake in this original version that was found and corrected by . The theorem considered by Grunwald and Wang was more general than the one stated above as they discussed the existence of cyclic extensions with certain local properties, and the statement about ''n''th powers is a consequence of this. History , a student of Helmut Hasse, gave an incorrect proof of the erroneous statement that an element in a number field is an ''n''th power if it is an ''n''th power locally almost everywhere. gave anothe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal completed the next year, after a regulatory review. Thus, Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ... Well-known products include the '' Methods in Enzymology'' series and encyclopedias such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
