In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a theorem about polynomials over the integers, or, more generally, over a unique factorization...

Gauss's lemma can mean any of several mathematical lemmas named after Carl Friedrich Gauss: Gauss's lemma (polynomials), the greatest common divisor of...

Brahmagupta polynomials Caloric polynomial Charlier polynomials Chebyshev polynomials Chihara–Ismail polynomials Cyclotomic polynomials Dickson polynomial Ehrhart...

or Gaussian method Gauss–Jordan elimination Gauss–Seidel method Gauss's cyclotomic formula Gauss's lemma in relation to polynomials Gaussian binomial coefficient...

Ewald, Gauss's son-in-law, and Wolfgang Sartorius von Waltershausen, Gauss's close friend and biographer, gave eulogies at his funeral. Gauss was a successful...

mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Bézout's identity—Let...

is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of...

theory) Zassenhaus lemma Gauss's lemma (polynomials) Schwartz–Zippel lemma Artin–Rees lemma Hensel's lemma (commutative rings) Nakayama lemma Noether's normalization...

non-constant polynomials are exactly the polynomials that are non-invertible and non-zero. Another definition is frequently used, saying that a polynomial is irreducible...

Taylor, Paul (2 June 2007), Gauss's second proof of the fundamental theorem of algebra – English translation of Gauss's second proof. van der Waerden...

polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. The original use of interpolation polynomials was...

Poincaré lemma helps in finding such potentials when the magnetic field is "well-behaved" (i.e., when the magnetic field is not due to a monopole), Gauss's law...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

symmetric polynomials Fundamental theorem of topos theory Fundamental theorem of ultraproducts Fundamental theorem of vector analysis Carl Friedrich Gauss referred...

the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that...

primitive part by the inverse of this unit. Gauss's lemma implies that the product of two primitive polynomials is primitive. It follows that primpart ⁡...

multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with...

A polynomial is primitive if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial. Gauss's lemma for polynomials states...

primitive part. Gauss proved that the product of two primitive polynomials is also primitive (Gauss's lemma). This implies that a primitive polynomial is irreducible...

^{7}-x^{6}-x^{5}+x^{2}+x+1.\end{aligned}}} The cyclotomic polynomials are monic polynomials with integer coefficients that are irreducible over the field...

positive integers n. It is hard to approximate it by a single smooth polynomial. The Gauss–Kuzmin–Wirsing operator G {\displaystyle G} acts on functions f...

follow from cases 1) - 8). Gauss needed, and was able to prove, a lemma similar to the one Legendre needed: Gauss's Lemma. If p is a prime congruent to...

for the first time the fundamental theorem of arithmetic. Article 16 of Gauss's Disquisitiones Arithmeticae is an early modern statement and proof employing...

case of Chinese remainder theorem for polynomials is Lagrange interpolation. For this, consider k monic polynomials of degree one: P i ( X ) = X − x i ...

the product of lower-degree polynomials with integer coefficients. This criterion is applicable only to monic polynomials. However, unlike other commonly...

the fastest algorithms today for computer computation of real roots of polynomials (see real-root isolation). Descartes himself used the transformation...

In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct...

of transcendence degree. Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step. Burau, Werner (2008), "Lueroth (or Lüroth)...

is actually much later than Fermat's; it first appears in section 1 of Gauss's Disquisitiones Arithmeticae. Fermat's little theorem is a consequence of...

{g}}]} denote the algebra of C {\displaystyle \mathbb {C} } -valued polynomials on g {\displaystyle {\mathfrak {g}}} (exactly the same argument works...