|
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] |
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 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 Diophantin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic 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, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual 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. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defined dually as an element of ''S'' that is not greater than any other element in ''S''. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S of a preordered set is an element of S which is greater than or equal to any other element of S, and the minimum of S is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. As an ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian 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. 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 and the ''n''-th ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unramified
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. See also branch point. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F'' "). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Class Field
In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonically isomorphic to the ideal class group of ''K'' using Frobenius elements for prime ideals in ''K''. In this context, the Hilbert class field of ''K'' is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every real embedding of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E''). Examples *If the ring of integers of ''K'' is a unique factorization domain, in particular if K = \mathbb , then ''K'' is its own Hilbert class field. *Let K = \mathbb(\sqrt) of discriminant -15. The field L = \mathbb(\sqrt, \sqrt) has discriminant 225=(-15)^2 and so is an everywhere unramified extension of ''K'', and it is abelian. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
