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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 factorization ...
Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity, [4]: §§69–71 Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + i in Z[i], the ring of Gaussian integers.
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.
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.
Mathematical analysis. Nonstandard analysis. v. t. e. In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, [ 1 ] is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
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