Gauss's Lemma (other)
Gauss's lemma can mean any of several mathematical lemmas named after Carl Friedrich Gauss: * Gauss's lemma (polynomials), the greatest common divisor of the coefficients is a multiplicative function * Gauss's lemma (number theory), condition under which an integer is a quadratic residue * Gauss's lemma (Riemannian geometry), theorem in manifold theory * A generalization of Euclid's lemma is sometimes called Gauss's lemma See also * List of topics named after Carl Friedrich Gauss Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...

Gauss's Lemma (polynomials)
In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statementThis theorem is called a lemma for historical reasons. about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is ''primitive'' if it has 1 as a greatest common divisor of its coefficients). A corollary of Gauss's lemma, sometimes also called ''Gauss's lemma'', is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a un ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Gauss's Lemma (number Theory)
Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity. It made its first appearance in Carl Friedrich Gauss's third proof (1808) of quadratic reciprocity and he proved it again in his fifth proof (1818). Statement of the lemma For any odd prime let be an integer that is coprime to . Consider the integers :a, 2a, 3a, \dots, \fraca and their least positive residues modulo . These residues are all distinct, so there are ( of them. Let be the number of these residues that are greater than . Then :\left(\frac\right) = (-1)^n, where \left(\frac\right) is the Legendre symbol. Example Taking = 11 and = 7, the relevant sequence of integers is : 7, 14, 21, 28, 35. After reduction modulo 11, this sequence becomes : 7, 3, 10, 6, 2. Three of these integers are larger than 11/2 (namely 6, 7 and 10), so = 3. Corresp ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Gauss's Lemma (Riemannian Geometry)
In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let ''M'' be a Riemannian manifold, equipped with its Levi-Civita connection, and ''p'' a point of ''M''. The exponential map is a mapping from the tangent space at ''p'' to ''M'': :\mathrm : T_pM \to M which is a diffeomorphism in a neighborhood of zero. Gauss' lemma asserts that the image of a sphere of sufficiently small radius in ''T''p''M'' under the exponential map is perpendicular to all geodesics originating at ''p''. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and normal coordinates. Introduction We define the exponential map at p\in M by : \exp_p: T_pM\supset B_(0) \longrightarrow M,\quad vt \longmapsto \gamma_(t), where \gamma_ is the unique geodesic with \gamma_(0)= ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Euclid's Lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, . If the premise of the lemma does not hold, i.e., is a composite number, its consequent may be either true or false. For example, in the case of , , , composite number 10 divides , but 10 divides neither 4 nor 15. This property is the key in the proof of the fundamental theorem of arithmetic. It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's Lemma shows that in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains. Formulations Euclid's lemma is commonly used in the following equivalent form: Euclid's lemma can be generalized as follows from prime numbers to any integers. This is a ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]