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Genus Character
In number theory, a genus character of a quadratic number field ''K'' is a character of the genus group of ''K''. In other words, it is a real character of the narrow class group of ''K''. Reinterpreting this using the Artin map, the collection of genus characters can also be thought of as the unramified real characters of the absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ... of ''K'' (i.e. the characters that factor through the Galois group of the genus field of ''K''). References *Chapter II of * *Section 12.5 of *Section 2.3 of Algebraic number theory {{numtheory-stub ... [...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] |
Quadratic Number Field
In algebraic number theory, a quadratic field is an algebraic number field of 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 subfield of the field of the s. Quadratic fields have been studied in great depth, initially as part of the theory of s. There remain some unsolve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus Group
In algebraic number theory, the genus field ''Γ(K)'' of an algebraic number field ''K'' is the maximal abelian extension of ''K'' which is obtained by composing an absolutely abelian field with ''K'' and which is unramified at all finite primes of ''K''. The genus number of ''K'' is the degree 'Γ(K)'':''K''and the genus group is the Galois group of ''Γ(K)'' over ''K''. If ''K'' is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of ''K'' unramified at all finite primes: this definition was used by Leopoldt and Hasse. If ''K''=Q() (''m'' squarefree) is a quadratic field of discriminant ''D'', the genus field of ''K'' is a composite of quadratic fields. Let ''p''''i'' run over the prime factors of ''D''. For each such prime ''p'', define ''p''∗ as follows: : p^* = \pm p \equiv 1 \pmod 4 \text p \text ; : 2^* = -4, 8, -8 \text m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . Then the genus field is the composite K( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narrow Class Group
In algebraic number theory, the narrow class group of a number field ''K'' is a refinement of the class group of ''K'' that takes into account some information about embeddings of ''K'' into the field of real numbers. Formal definition Suppose that ''K'' is a finite extension of Q. Recall that the ordinary class group of ''K'' is defined as the quotient :C_K = I_K / P_K,\,\! where ''I''''K'' is the group of fractional ideals of ''K'', and ''P''''K'' is the subgroup of principal fractional ideals of ''K'', that is, ideals of the form ''aO''''K'' where ''a'' is an element of ''K''. The narrow class group is defined to be the quotient :C_K^+ = I_K / P_K^+, where now ''P''''K''+ is the group of totally positive principal fractional ideals of ''K''; that is, ideals of the form ''aO''''K'' where ''a'' is an element of ''K'' such that σ(''a'') is ''positive'' for every embedding :\sigma : K \to \mathbb. Uses The narrow class group features prominently in the theory of repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Map
The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem. Statement Let L/K be a Galois extension of global fields and C_L stand for the idèle class group of L. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol mapNeukirch (1999) p.391 : \theta: C_K/ \to \operatorname(L/K)^, where \text denotes the abelianization of a group, and \operatorname(L/K) is the Galois group of L over K. The map \theta is defined by assembling the maps called the local Art ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Galois Group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' that fix ''K''. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When ''K'' is a perfect field, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) Examples * The absolute Galois group of an algebraically closed field is trivial. * The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R, and its degree over R is ''C:Rnbsp;= 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group of profinite integers :: \hat = \varprojlim \mathbf/n\mathbf. :(For the notation, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus Field
In algebraic number theory, the genus field ''Γ(K)'' of an algebraic number field ''K'' is the maximal abelian extension of ''K'' which is obtained by composing an absolutely abelian field with ''K'' and which is unramified at all finite primes of ''K''. The genus number of ''K'' is the degree 'Γ(K)'':''K''and the genus group is the Galois group of ''Γ(K)'' over ''K''. If ''K'' is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of ''K'' unramified at all finite primes: this definition was used by Leopoldt and Hasse. If ''K''=Q() (''m'' squarefree) is a quadratic field of discriminant ''D'', the genus field of ''K'' is a composite of quadratic fields. Let ''p''''i'' run over the prime factors of ''D''. For each such prime ''p'', define ''p''∗ as follows: : p^* = \pm p \equiv 1 \pmod 4 \text p \text ; : 2^* = -4, 8, -8 \text m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . Then the genus field is the composite K( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
