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In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".
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.
The two first subsections, are proofs of the generalized version of Euclid's lemma, namely that: if n divides ab and is coprime with a then it divides b. The original Euclid's lemma follows immediately, since, if n is prime then it divides a or does not divide a in which case it is coprime with a so per the generalized version it divides b.
Burnside's lemma also known as the Cauchy–Frobenius lemma; Frattini's lemma (finite groups) Goursat's lemma; Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma; Schur's lemma (representation theory) Zassenhaus lemma
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).
This interpretation of the Legendre symbol as the sign of a permutation can be extended to the Jacobi symbol ( a n ) , {\displaystyle \left({\frac {a}{n}}\right),} where a and n are relatively prime integers with odd n > 0: a is invertible mod n , so multiplication by a on Z / n Z is a permutation and a generalization of Zolotarev's lemma is ...
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
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology .