polynomials (see Polynomial greatest common divisor) and other commutative rings (see § In commutative rings below). The greatest common divisor (GCD) of integers...

In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor...

their greatest common divisor. Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and...

and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive. (A polynomial with...

the definition of Gröbner bases. Euclidean algorithm for polynomial greatest common divisor computation and Gaussian elimination of linear systems are...

the Euclidean algorithm for polynomials that computes a polynomial greatest common divisor of two polynomials. Here, "greatest" means "having a maximal degree"...

near John Day, Oregon, United States Greatest common divisor Binary GCD algorithm Polynomial greatest common divisor Lehmer's GCD algorithm Griffith College...

nonzero polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its...

account. Other applications of multi-modular arithmetic include polynomial greatest common divisor, Gröbner basis computation and cryptography. A residue numeral...

performing Euclidean polynomial division Ruffini's rule Euclidean domain Gröbner basis Greatest common divisor of two polynomials Archived at Ghostarchive...

1} is a greatest common divisor of the polynomial and its derivative. A square-free decomposition or square-free factorization of a polynomial is a factorization...

who proved it for polynomials, is the following theorem: Bézout's identity — Let a and b be integers with greatest common divisor d. Then there exist...

generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases...

those of f, and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of f and f′ has coefficients...

arithmetic. It is used by the most efficient implementations of polynomial greatest common divisor, exact linear algebra and Gröbner basis algorithms over the...

P} and Q {\displaystyle \textstyle Q} have a non-constant polynomial greatest common divisor R {\displaystyle \textstyle R} , then setting P = P 1 R {\displaystyle...

of a rational number and a polynomial with integer coefficients, which is primitive (that is, the greatest common divisor of the coefficients is 1), and...

performed within the same time bounds. The cost of a polynomial greatest common divisor between two polynomials of degree at most n can be taken as O(n2) operations...

needs of the simplifier. For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions...

reorganizing symbolic algebraic expressions, such as polynomial factorization, polynomial greatest common divisor calculation, expansion, separation into real...

approximant is via the extended Euclidean algorithm for the polynomial greatest common divisor. The relation R ( x ) = P ( x ) / Q ( x ) = T m + n ( x )...

or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers, the largest number that divides them both...

generated by the greatest common divisor of these minors. As one is working with polynomials with integer coefficients, this greatest common divisor is defined...

number of their intersections is valid, if their polynomial greatest common divisor is a constant. A meromorphic function is a ratio of two well-behaved...

{26}{11}}} . This simplification may be done by factoring out the greatest common divisor. Give the answer as an integer quotient and a remainder, so 26...

Polynomial greatest common divisor § Bézout's identity and extended GCD algorithm for details. Linear algebra is effective on a polynomial ring k [ x...

divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors...

primitive polynomial. A primitive polynomial is a polynomial over a unique factorization domain, such that 1 is a greatest common divisor of its coefficients...

if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer...

j)=n}\gcd(i,j)M(\alpha ,i)M(\beta ,j),} where "gcd" is greatest common divisor and "lcm" is least common multiple. More generally, M ( α β ⋯ γ , n ) = ∑ lcm...