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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 ...
This article is about Gauss's lemma in number theory. For other uses, see Gauss's lemma. Gauss's lemmain number theorygives 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.
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.
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.
By Gauss's lemma, if the equation is reducible, one can suppose that the factors have integer coefficients. Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form q x − p {\displaystyle qx-p} with q and p ...
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