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List Of Algebraic Number Theory Topics
This is a list of algebraic number theory topics. Basic topics These topics are basic to the field, either as prototypical examples, or as basic objects of study. *Algebraic number field **Gaussian integer, Gaussian rational **Quadratic field **Cyclotomic field **Cubic field ** Biquadratic field *Quadratic reciprocity *Ideal class group *Dirichlet's unit theorem *Discriminant of an algebraic number field *Ramification (mathematics) *Root of unity *Gaussian period Important problems *Fermat's Last Theorem * Class number problem for imaginary quadratic fields **Stark–Heegner theorem ***Heegner number *Langlands program General aspects *Different ideal *Dedekind domain *Splitting of prime ideals in Galois extensions *Decomposition group *Inertia group *Frobenius automorphism *Chebotarev's density theorem *Totally real field *Local field ** ''p''-adic number ** ''p''-adic analysis *Adele ring * Idele group *Idele class group *Adelic algebraic group *Global field *Hasse principle * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory 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 integers ''x'' and ''y'' such that their sum, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stark–Heegner Theorem
In number theory, the Baker–Heegner–Stark theorem states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number. Let Q denote the set of rational numbers, and let ''d'' be a non-square integer. Then Q() is a finite extension of Q of degree 2, called a quadratic extension. The class number of Q() is the number of equivalence classes of ideals of the ring of integers of Q(), where two ideals ''I'' and ''J'' are equivalent if and only if there exist principal ideals (''a'') and (''b'') such that (''a'')''I'' = (''b'')''J''. Thus, the ring of integers of Q() is a principal ideal domain (and hence a unique factorization domain) if and only if the class number of Q() is equal to 1. The Baker–Heegner–Stark theorem can then be stated as follows: :If ''d'' 0 for which Q() has class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Analysis
In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of locally compact groups. The usual meaning taken for ''p''-adic analysis is the theory of ''p''-adic-valued functions on spaces of interest. Applications of ''p''-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of ''p''-adic functional analysis and spectral theory. In many ways ''p''-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of ''p''-adic numbers is much simpler. Topological vector spaces over ''p''-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Real Field
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polynomial ''P'', all of the roots of ''P'' being real; or that the tensor product algebra of ''F'' with the real field, over Q, is isomorphic to a tensor power of R. For example, quadratic fields ''F'' of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial ''P'' irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will ''not'' be totally real, although it is a field of real numbers. The totally real number fields play a significant special role in algebraic number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebotarev's Density Theorem
Chebotarev's 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'' among ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Automorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general. Definition Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism ''F'' is defined by :F(r) = r^p for all ''r'' in ''R''. It respects the multiplication of ''R'': :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients :\frac, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inertia Group
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. Ramification theory of valuations In mathematics, the ramification theory of valuations studies the set of extensions of a valuation ''v'' of a field ''K'' to an extension ''L'' of ''K''. It is a generalization of the ramification theory of Dedekind domains. The structure of the set of extensions is known better when ''L''/''K'' is Galois. Decomposition group and inertia group Let (''K'', ''v'') be a valued field and let ''L'' be a finite Galois extension of ''K''. Let ''Sv'' be the set of equivalence classes of extensions of ''v'' to ''L'' and let ''G'' be the Galois group of ''L'' over ''K''. Then ''G'' acts on ''Sv'' by σ 'w''nbsp;= 'w'' ∘ σ(i.e. ''w'' is a representative of the equivalence class 'w''nbsp;∈ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomposition Group
In mathematics, the interplay between the Galois group ''G'' of a Galois extension ''L'' of a number field ''K'', and the way the prime ideals ''P'' of the ring of integers ''O''''K'' factorise as products of prime ideals of ''O''''L'', provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of ''G'' need be considered, rather than two. This was certainly familiar before Hilbert. Definitions Let ''L''/''K'' be a finite extension of number fields, and let ''OK'' and ''OL'' be the corresponding ring of integers of ''K'' and ''L'', respectively, which are defined to be the integral closure of the integers Z in the field in question. : \begin O_K & \hookrightarrow & O_L \\ \downarrow & & \downarrow \\ K & \hookrightarrow & L \end Fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Splitting Of Prime Ideals In Galois Extensions
In mathematics, the interplay between the Galois group ''G'' of a Galois extension ''L'' of a number field ''K'', and the way the prime ideals ''P'' of the ring of integers ''O''''K'' factorise as products of prime ideals of ''O''''L'', provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of ''G'' need be considered, rather than two. This was certainly familiar before Hilbert. Definitions Let ''L''/''K'' be a finite extension of number fields, and let ''OK'' and ''OL'' be the corresponding ring of integers of ''K'' and ''L'', respectively, which are defined to be the integral closure of the integers Z in the field in question. : \begin O_K & \hookrightarrow & O_L \\ \downarrow & & \downarrow \\ K & \hookrightarrow & L \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
