|
Shintani's Unit Theorem
In mathematics, Shintani's unit theorem introduced by is a refinement of Dirichlet's unit theorem and states that a subgroup of finite index of the totally positive units of a number field has a fundamental domain given by a rational polyhedric cone in the Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ... of the field . References * * *{{citation, mr=0633664, last=Shintani, first= Takuro , chapter=A remark on zeta functions of algebraic number fields, title= Automorphic forms, representation theory and arithmetic (Bombay, 1979), pages= 255–260 , series=Tata Inst. Fund. Res. Studies in Math., volume= 10, publisher= Tata Inst. Fundamental Res., place= Bombay, year= 1981, isbn=3-540-10697-9 External links Mathematical picturesby Paul Gunnells Theorems in algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet's Unit Theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a positive real number that determines how "dense" the units are. The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to where is the ''number of real embeddings'' and the ''number of conjugate pairs of complex embeddings'' of . This characterisation of and is based on the idea that there will be as many ways to embed in the complex number field as the degree n = : \mathbb/math>; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that Note that if is Galois over \mathbb then either or . Other ways of determining and are * use the primitive element theorem to write K = \mathbb(\alpha), and then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Space (number Field)
In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field. If ''K'' is a number field of Degree of a number field, degree ''d'' then there are ''d'' distinct embeddings of ''K'' into C. We let ''K''C be the image of ''K'' in the product C''d'', considered as equipped with the usual Hermitian inner product. If ''c'' denotes complex conjugation, let ''K''R denote the subspace of ''K''C fixed by ''c'', equipped with a scalar product. This is the Minkowski space of ''K''. See also * Geometry of numbers Footnotes References * {{cite book , first=Jürgen , last=Neukirch , authorlink=Jürgen Neukirch , title=Algebraic Number Theory , volume=322 , series=Grundlehren der Mathematischen Wissenschaften , publisher=Springer-Verlag , year=1999 , isbn=978-3-540-65399-8 , zbl=0956.11021 , mr=1697859 Algebraic number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
