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Teichmüller Character
In number theory, the Teichmüller character ω (at a prime ''p'') is a character of (Z/''q''Z)×, where q = p if p is odd and q = 4 if p = 2, taking values in the roots of unity of the ''p''-adic integers. It was introduced by Oswald Teichmüller. Identifying the roots of unity in the ''p''-adic integers with the corresponding ones in the complex numbers, ω can be considered as a usual Dirichlet character of conductor ''q''. More generally, given a complete discrete valuation ring ''O'' whose residue field ''k'' is perfect of characteristic ''p'', there is a unique multiplicative section of the natural surjection . The image of an element under this map is called its Teichmüller representative. The restriction of ω to ''k''× is called the Teichmüller character. Definition If ''x'' is a ''p''-adic integer, then \omega(x) is the unique solution of \omega(x)^p = \omega(x) that is congruent to ''x'' mod ''p''. It can also be defined by :\omega(x)=\lim_ x^ The multiplica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character Theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations. Applications Characters of irreducible representations encode many important propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oswald Teichmüller
Paul Julius Oswald Teichmüller (; 18 June 1913 – 11 September 1943) was a German mathematician who made contributions to complex analysis. He introduced quasiconformal mappings and differential geometric methods into the study of Riemann surfaces. Teichmüller spaces are named after him. He was a supporter of the Nazi regime. Born in Nordhausen, Teichmüller attended the University of Göttingen, where he graduated in 1935 under the supervision of Helmut Hasse. His doctoral dissertation was on operator theory, though this was his only work on functional analysis. His next few papers were algebraic, but he switched his focus to complex analysis after attending lectures given by Rolf Nevanlinna. In 1937, he moved to the University of Berlin to work with Ludwig Bieberbach. Bieberbach was the editor of ''Deutsche Mathematik'' and much of Teichmüller's work was published in the journal, which made his papers hard to find in modern libraries before the release of his collected works ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1) \chi(ab) = \chi(a)\chi(b); i.e. \chi is completely multiplicative. :2) \chi(a) \begin =0 &\text\; \gcd(a,m)>1\\ \ne 0&\text\;\gcd(a,m)=1. \end (gcd is the greatest common divisor) :3) \chi(a + m) = \chi(a); i.e. \chi is periodic with period m. The simplest possible character, called the principal character, usually denoted \chi_0, (see Notation below) exists for all moduli: : \chi_0(a)= \begin 0 &\text\; \gcd(a,m)>1\\ 1 &\text\;\gcd(a,m)=1. \end The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions. Notation \phi(n) is Euler's totient function. \zeta_n is a complex primitive n-th root of unity: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete (topology)
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots in a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d\left(x_m, x_n\right) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: :#Every |
Discrete Valuation Ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' is a local principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local Dedekind domain and not a field. # ''R'' is a Noetherian local domain whose maximal ideal is principal, and not a field.https://mathoverflow.net/a/155639/114772 # ''R'' is an integrally closed Noetherian local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element ( up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Field Of A Valuation
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : which satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(''a'') ≠ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Field
In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is separable. * Every algebraic extension of ''k'' is separable. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , every element of ''k'' is a ''p''th power. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , the Frobenius endomorphism is an automorphism of ''k''. * The separable closure of ''k'' is algebraically closed. * Every reduced commutative ''k''-algebra ''A'' is a separable algebra; i.e., A \otimes_k F is reduced for every field extension ''F''/''k''. (see below) Otherwise, ''k'' is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Section (category Theory)
In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f: X\to Y and g: Y\to X are morphisms whose composition f \circ g: Y\to Y is the identity morphism on Y, then g is a section of f, and f is a retraction of g. Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative). In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if f: X\to Y is a split epimorphism with split monomorphism g: Y\to X, then X is isomorphic to the direct sum of Y and the kernel of f. The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work. Properties * A section that is also an epimorphism is an isomorphism. Dually a retraction that is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hensel Lifting
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to a unique root modulo any higher power of . More generally, if a polynomial factors modulo into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of (the case of roots corresponds to the case of degree for one of the factors). By passing to the "limit" (in fact this is an inverse limit) when the power of tends to infinity, it follows that a root or a factorization modulo can be lifted to a root or a factorization over the -adic integers. These results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where is replaced by an ideal, and "coprime polynomials" means "polynomials that generate an ideal containing ". Hense ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
