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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 ...
The proof of the n th-power lemma uses the same ideas that were used in the proof of the quadratic lemma. The existence of the integers π(i) and b(i), and their uniqueness (mod m) and (mod n), respectively, come from the fact that Aμ is a representative set. Assume that π(i) = π(j) = p, i.e.
Gauss needed, and was able to prove, [18] a lemma similar to the one Legendre needed: Gauss's Lemma. If p is a prime congruent to 1 modulo 8 then there exists an odd prime q such that: < + = The proof of quadratic reciprocity uses complete induction.
Proof using Gauss's lemma [ edit ] Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma ; this does not alter the set of rational roots and only strengthens the divisibility ...
This is an accepted version of this page This is the latest accepted revision, reviewed on 27 September 2024. German mathematician, astronomer, geodesist, and physicist (1777–1855) "Gauss" redirects here. For other uses, see Gauss (disambiguation). Carl Friedrich Gauss Portrait by Christian Albrecht Jensen, 1840 (copy from Gottlieb Biermann, 1887) Born Johann Carl Friedrich Gauss (1777-04-30 ...
Fundamental theorem of algebra. The fundamental theorem of algebra, also called d'Alembert's theorem[1] or the d'Alembert–Gauss theorem, [2] states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex ...
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements. If two numbers by multiplying one another make somenumber, and any prime number measure the product, it willalso measure one of the original numbers. — Euclid, Elements Book VII, Proposition 30.
Gauss's lemma. 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 ...