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In algebra, Gauss's lemma, [ 1 ] named after Carl Friedrich Gauss, is a theorem [ note 1 ] 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 ...
7, 3, 10, 6, 2. Three of these integers are larger than 11/2 (namely 6, 7 and 10), so n = 3. Correspondingly Gauss's lemma predicts that = = This is indeed correct, because 7 is not a quadratic residue modulo 11. The above sequence of residues 7, 3, 10, 6, 2. may also be written
Rational root theorem. In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer coefficients and . Solutions of the equation are also called roots or zeros of the polynomial on the left side.
Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125–146 and 262 of Disquisitiones Arithmeticae in 1801.. 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.
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
In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible, called the Levi-Civita connection or (pseudo-)Riemannian connection of the given metric.
Created Date: 8/30/2012 4:52:52 PM
Gauss's lemma for polynomials states that the product of primitive polynomials (with coefficients in the same unique factorization domain) also is primitive. This implies that the content and the primitive part of the product of two polynomials are, respectively, the product of the contents and the product of the primitive parts.