|
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) ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Cubic Field
In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three. Definition If ''K'' is a field extension of the rational numbers Q of degree 'K'':Qnbsp;= 3, then ''K'' is called a cubic field. Any such field is isomorphic to a field of the form :\mathbf (f(x)) where ''f'' is an irreducible cubic polynomial with coefficients in Q. If ''f'' has three real roots, then ''K'' is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, ''f'' has a non-real root, then ''K'' is called a complex cubic field. A cubic field ''K'' is called a cyclic cubic field if it contains all three roots of its generating polynomial ''f''. Equivalently, ''K'' is a cyclic cubic field if it is a Galois extension of Q, in which case its Galois group over Q is cyclic of order three. This can only happen if ''K'' is totally real. It is a rare occurrence in the sense that if the set of cubic fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Unit (number Theory)
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units. Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. ). Real quadratic fields For the real quadratic field K=\mathbf(\sqrt) (with ''d'' square-free), the fundamental unit ε is commonly normalized so that (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Quadratic 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 unsolved prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Number Formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. General statement of the class number formula We start with the following data: * is a number field. * , where denotes the number of real embeddings of , and is the number of complex embeddings of . * is the Dedekind zeta function of . * is the class number, the number of elements in the ideal class group of . * is the regulator of . * is the number of roots of unity contained in . * is the discriminant of the extension . Then: :Theorem (Class Number Formula). converges absolutely for and extends to a meromorphic function defined for all complex with only one simple pole at , with residue :: \lim_ (s-1) \zeta_K(s) = \frac This is the most general "class number formula". In particular cases, for example when is a cyclotomic extension of , there are particular and more refined class number formulas. Proof The i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Number (number Theory)
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of . The order of the group, which is finite, is called the class number of . The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. History and origin of the ideal class group Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as pu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denoted ''v'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object ( algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S-unit
In mathematics, in the field of algebraic number theory, an ''S''-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units. Definition Let ''K'' be a number field with ring of integers ''R''. Let ''S'' be a finite set of prime ideals of ''R''. An element ''x'' of ''K'' is an ''S''-unit if the principal fractional ideal (''x'') is a product of primes in ''S'' (to positive or negative powers). For the ring of rational integers Z one may take ''S'' to be a finite set of prime numbers and define an ''S''-unit to be a rational number whose numerator and denominator are divisible only by the primes in ''S''. Properties The ''S''-units form a multiplicative group containing the units of ''R''. Dirichlet's unit theorem holds for ''S''-units: the group of ''S''-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to ''r'' + ''s'', where ''r'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a founding member of the Bourbaki group. Life His father, Abel Chevalley, was a French diplomat who, jointly with his wife Marguerite Chevalley née Sabatier, wrote ''The Concise Oxford French Dictionary''. Chevalley graduated from the École Normale Supérieure in 1929, where he studied under Émile Picard. He then spent time at the University of Hamburg, studying under Emil Artin and at the University of Marburg, studying under Helmut Hasse. In Germany, Chevalley discovered Japanese mathematics in the person of Shokichi Iyanaga. Chevalley was awarded a doctorate in 1933 from the University of Paris for a thesis on class field theory. When World War II broke out, Chevalley was at Princeton University. After reporting to the French Embass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
