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Hilbert Symbol
In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of real number, reals or p-adic numbers. It is related to reciprocity law (mathematics), reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields. The Hilbert symbol has been generalized to higher local fields. Quadratic Hilbert symbol Over a local field K with multiplicative group of non-zero elements K^\times, the quadratic Hilbert symbol is the function (mathematics), function K^\times\times K^\times\to\ defined by :(a,b)=\begin+1,&\mboxz^2=ax^2+by^2\mbox(x,y,z)\in K^3;\\-1,&\mbox\end Equivalently, (a, b) = 1 if and only if b is equal to the Field norm, norm of an element of the quadr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Conjecture (K-theory)
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' with coefficients in Z/2Z. It was proved by . Statement Let ''F'' be a field of characteristic different from 2. Then there is an isomorphism :K_n^M(F)/2 \cong H_^n(F, \mathbb/2\mathbb) for all ''n'' ≥ 0, where ''KM'' denotes the Milnor ring. About the proof The proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Alexander Merkurjev, Andrei Suslin, Markus Rost, Fabien Morel, Eric Friedlander, and others, including the newly minted theory of motivic cohomology (a kind of substitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra. Generalizations The analogue of this result for primes other than 2 was known as the Bloch–Kato conjecture. Work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Residue Symbol
In algebraic number theory the ''n''-th power residue symbol (for an integer ''n'' > 2) is a generalization of the (quadratic) Legendre symbol to ''n''-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher reciprocity laws. Background and notation Let ''k'' be an algebraic number field with ring of integers \mathcal_k that contains a primitive ''n''-th root of unity \zeta_n. Let \mathfrak \subset \mathcal_k be a prime ideal and assume that ''n'' and \mathfrak are coprime (i.e. n \not \in \mathfrak.) The norm of \mathfrak is defined as the cardinality of the residue class ring (note that since \mathfrak is prime the residue class ring is a finite field): :\mathrm \mathfrak := , \mathcal_k / \mathfrak, . An analogue of Fermat's theorem holds in \mathcal_k. If \alpha \in \mathcal_k - \mathfrak, then :\alpha^\equiv 1 \bmod. And finally, suppose \mathrm \mathfrak \equiv 1 \bmod. These facts imply that :\alpha^\equiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Reciprocity Law
The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit .... The term "reciprocity law (mathematics), reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Gotthold Eisenstein, Eisenstein and Ernst Kummer, Kummer to David Hilbert, Hilbert's product formula for the Hilbert symbol, norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem. Statement Let L/K be a Galois extension of global fields and C_L stand for the Adelic algebraic group, idèle class group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
U-invariant
In mathematics, the universal invariant or ''u''-invariant of a field describes the structure of quadratic forms over the field. The universal invariant ''u''(''F'') of a field ''F'' is the largest dimension of an anisotropic quadratic space over ''F'', or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that ''u'' is the smallest number such that every form of dimension greater than ''u'' is isotropic, or that every form of dimension at least ''u'' is universal. Examples * For the complex numbers, ''u''(C) = 1. * If ''F'' is quadratically closed then ''u''(''F'') = 1. * The function field of an algebraic curve over an algebraically closed field has ''u'' ≤ 2; this follows from Tsen's theorem that such a field is quasi-algebraically closed.Lam (2005) p.376 * If ''F'' is a non-real global or local field, or more generall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x^2\equiv a \bmod p for an odd prime p; that is, to determine the "perfect squares" modulo p. However, this is a non-constructive result: it gives no help at all for finding a ''specific'' solution; for this, other methods are required. For example, in the case p\equiv 3 \bmod 4 using Euler's criterion one can give an explicit formula for the "square roots" modulo p of a quadratic residue a, namely, :\pm a^ indeed, :\left (\pm a^ \right )^2=a^=a\cdot a^\equiv a\left(\frac\right)=a \bmod p. This formula only works if it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue (''non-residue'') is −1. Its value at zero is 0. The Legendre symbol was introduced by Adrien-Marie Legendre in 1797 or 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol. Definition Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is modular arithmetic, congruent to a square number, perfect square modulo p and is a quadratic nonresidue modulo p otherwise. The Legendre symbol is a function of a a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Space
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 of elements from X of 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(x_m, x_n) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: #Every Cauchy se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number Field
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
