Search results
Results from the WOW.Com Content Network
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 factorization ...
Gauss's lemma is used in many, [3]: Ch. 1 [3]: 9 but by no means all, of the known proofs of quadratic reciprocity. For example, Gotthold Eisenstein [3]: 236 used Gauss's lemma to prove that if p is an odd prime then
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'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.
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 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.
Lemma (mathematics) In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem". [3][4] In many cases, a lemma derives its importance from the theorem it ...
Normal coordinates. In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection ...