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Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a Field extension, subfield of the field of the real numbers. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important. Ring of integers Discriminant For a nonzero square free integer , the Discriminant of an algebraic number field, discriminant of the quadratic field is |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebotarev Density Theorem
The Chebotarev density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of ''K''. There are only finitely many patterns of splitting that may occur. Although the full description of the splitting of every prime ''p'' in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes ''p'' less than a large integer ''N'', tends to a certain limit as ''N'' goes to infinity. It was proved by Nikolai Chebotaryov in his thesis in 1922, published in . A special case that is easier to state says that if ''K'' is an algebraic number field which is a Galois extension of \mathbb of degree ''n'', then the prime numbers that completely split in ''K'' have density :1/''n'' am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal Domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some authors such as Bourbaki refer to PIDs as principal rings. Principal ideal domains are mathematical objects that behave like the integers, with respect to divisibility: any element of a PID has a unique factorization into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form , etc. Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (ring Theory)
In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that #''A'' is a finite-dimensional algebra over the field \mathbb of rational numbers #\mathcal spans ''A'' over \mathbb, and #\mathcal is a \mathbb- lattice in ''A''. The last two conditions can be stated in less formal terms: Additively, \mathcal is a free abelian group generated by a basis for ''A'' over \mathbb. More generally for ''R'' an integral domain with fraction field ''K'', an ''R''-order in a finite-dimensional ''K''-algebra ''A'' is a subring \mathcal of ''A'' which is a full ''R''-lattice; i.e. is a finite ''R''-module with the property that ''\mathcal\otimes_RK=A''. When ''A'' is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conductor-discriminant Formula
In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by for abelian extensions and by for Galois extensions, is a formula calculating the relative discriminant of a finite Galois extension L/K of local or global fields from the Artin conductors of the irreducible characters \mathrm(G) of the Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ... G = G(L/K). Statement Let L/K be a finite Galois extension of global fields with Galois group G. Then the discriminant equals :: \mathfrak_ = \prod_\mathfrak(\chi)^, where \mathfrak(\chi) equals the global Artin conductor of \chi. Example Let L = \mathbf(\zeta_)/\mathbf be a cyclotomic extension of the rationals. The Galois group G equals (\mathbf/p^n)^\times. Because ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conductor (class Field Theory)
In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map. Local conductor Let ''L''/''K'' be a finite abelian extension of non-archimedean local fields. The conductor of ''L''/''K'', denoted \mathfrak(L/K), is the smallest non-negative integer ''n'' such that the higher unit group :U^ = 1 + \mathfrak_K^n = \left\ is contained in ''N''''L''/''K''(''L''×), where ''N''''L''/''K'' is field norm map and \mathfrak_K is the maximal ideal of ''K''. Equivalently, ''n'' is the smallest integer such that the local Artin map is trivial on U_K^. Sometimes, the conductor is defined as \mathfrak_K^n where ''n'' is as above. The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero, and it is tamely ramified if, and only if, the conducto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramification (mathematics)
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local pattern: if we exc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of root of unity, roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods. History As the name suggests, the periods were introduced by Carl Friedrich Gauss, Gauss and were the basis for his theory of compass and straightedge construction. For example, the construction of the heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which : 2 \cos \left(\frac\right) = \zeta + \zeta^ \, is an example involving the seventeenth root of unity : \zeta = \exp \left(\frac\right). General definition Given an integer ''n'' > 1, let ''H'' be any subgroup of the multiplicative group : G = (\mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Of A Subgroup
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left Coset, cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or [G:H] or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same cardinality, size as ''H'', the index is related to the order (group theory), orders of the two groups by the formula :, G, = , G:H, , H, (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index , G:H, measures the "relative sizes" of ''G'' and ''H''. For example, let G = \Z be the group of integers under addition, and let H = 2\Z be the subgroup consisting of the Parity (mathematics), even integers. Then 2\Z has two cosets in \Z, namely the set of even integers and the set of odd integers, so the index , \Z:2\Z, is 2. More generally, , \Z:n\Z, = n for any positive integer ''n''. When ''G'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying root of a function, roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is by definition ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, nth root, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n)—and more precisely, because of the failure of unique factorization in their rings of integers—that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For n \geq 1, let :\zeta_n=e^\in\C. This is a primitive nth root of unity. Then the nth cyclotomic field is the field extension \mathbb(\zeta_n) of \mathbb generated by \zeta_n. Properties * The nth cyclotomic polynomial :: \Phi_n(x) = \prod_\stackrel\!\!\! \left(x-e^\right) = \prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of \zeta_n o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind–Kummer Theorem
In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure. It is named after Richard Dedekind who developed the theorem based on the work of Ernst Kummer. Statement for number fields Let K be a number field such that K = \Q(\alpha) for \alpha \in \mathcal O_K and let f be the minimal polynomial for \alpha over \Z /math>. For any prime p not dividing irreducible polynomials in \mathbb F_p /math>. Then (p) = p \mathcal O_K factors into prime ideals as (p) = \mathfrak p_1^ \cdots \mathfrak p_g^ such that N(\mathfrak p_i) = p^, where N is the ideal norm. Statement for Dedekind domains The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let \mathcal o be a Dedekind domain contained in its quotient field K, L/K a finite, separable field extension with L=Ktheta Theta (, ) uppercase Θ or ; lowercase θ or ; ''thē̂ta'' ; Modern: ''thī́ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
