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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] [Amazon] |
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] [Amazon] |
Artin Conductor
In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function. Local Artin conductors Suppose that ''L'' is a finite Galois extension of the local field ''K'', with Galois group ''G''. If \chi is a character of ''G'', then the Artin conductor of \chi is the number :f(\chi)=\sum_\frac(\chi(1)-\chi(G_i)) where ''G''''i'' is the ''i''-th ramification group (in lower numbering), of order ''g''''i'', and χ(''G''''i'') is the average value of \chi on ''G''''i''.Serre (1967) p.158 By a result of Artin, the local conductor is an integer.Serre (1967) p.159 Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial. In particular, if χ is unramified, then its Artin conductor is zero. Thus if ''L'' is unramified over ''K'', then the Artin conductors of all χ are zero. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Kronecker–Weber Theorem
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form modular arithmetic, (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian group, abelian can be expressed as a sum of root of unity, roots of unity with rational coefficients. For example, :\sqrt = e^ - e^ - e^ + e^, \sqrt = e^ - e^, and \sqrt = e^ - e^. The theorem is named after Leopold Kronecker and Heinrich Martin Weber. Field-theoretic formulation The Kronecker–Weber theorem can be stated in terms of field (mathematics), fields and field extensions. Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field. That is, whenever an algebraic number field has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ray Class Group
In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class fields. The term "ray class group" is a translation of the German term "Strahlklassengruppe". Here "Strahl" is German for ray, and often means the positive real line, which appears in the positivity conditions defining ray class groups. uses "Strahl" to mean a certain group of ideals defined using positivity conditions, and uses "Strahlklasse" to mean a coset of this group. There are two slightly different notions of what a ray class field is, as authors differ in how the infinite primes are treated. History Weber introduced ray class groups in 1897. Takagi proved the existence of the corresponding ray class fields in about 1920. Chevalley reformulated the definition of ray class groups in terms of ideles in 1933. Ray class fields using ideals If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |