Euclidean algorithm for polynomials

E646780

algorithm greatest common divisor algorithm

The Euclidean algorithm for polynomials is a procedure that repeatedly applies polynomial division to compute the greatest common divisor of two polynomials over a given field or ring.

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All labels observed (5)

Label	Occurrences
Euclidean algorithm	4
Bezout's identity	1
Euclidean algorithm for integers	1
Euclidean algorithm for polynomials canonical	1
extended Euclidean algorithm	1

How this entity was disambiguated

This entity first appeared as the object of triple T7174363 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Euclidean algorithm for polynomials
Context triple: [Berlekamp–Massey algorithm, relatedTo, Euclidean algorithm for polynomials]

A. Cantor–Zassenhaus algorithm
The Cantor–Zassenhaus algorithm is a probabilistic method used to factor polynomials over finite fields efficiently, widely employed in computational algebra and cryptography.
B. Berlekamp’s algorithm for factoring polynomials over finite fields
Berlekamp’s algorithm for factoring polynomials over finite fields is a foundational deterministic method in computational algebra that efficiently decomposes polynomials into irreducible factors over finite fields and underpins many modern algorithms in coding theory and cryptography.
C. Buchberger algorithm
The Buchberger algorithm is a fundamental procedure in computational algebra for computing Gröbner bases of polynomial ideals, enabling systematic solutions to systems of polynomial equations.
D. Berlekamp–Massey algorithm
The Berlekamp–Massey algorithm is a key algorithm in coding theory and cryptography used to efficiently determine the shortest linear feedback shift register that generates a given binary sequence.
E. Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Euclidean algorithm for polynomials
Target entity description: The Euclidean algorithm for polynomials is a procedure that repeatedly applies polynomial division to compute the greatest common divisor of two polynomials over a given field or ring.

A. Cantor–Zassenhaus algorithm
The Cantor–Zassenhaus algorithm is a probabilistic method used to factor polynomials over finite fields efficiently, widely employed in computational algebra and cryptography.
B. Berlekamp’s algorithm for factoring polynomials over finite fields
Berlekamp’s algorithm for factoring polynomials over finite fields is a foundational deterministic method in computational algebra that efficiently decomposes polynomials into irreducible factors over finite fields and underpins many modern algorithms in coding theory and cryptography.
C. Buchberger algorithm
The Buchberger algorithm is a fundamental procedure in computational algebra for computing Gröbner bases of polynomial ideals, enabling systematic solutions to systems of polynomial equations.
D. Berlekamp–Massey algorithm
The Berlekamp–Massey algorithm is a key algorithm in coding theory and cryptography used to efficiently determine the shortest linear feedback shift register that generates a given binary sequence.
E. Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	algorithm ⓘ greatest common divisor algorithm ⓘ
applicableOver	Euclidean domains ⓘ fields ⓘ principal ideal domains via embeddings ⓘ
basedOn	Euclidean algorithm for integers ⓘ linked to: Euclidean algorithm for polynomials
complexityDependsOn	cost of polynomial multiplication and division ⓘ degrees of the input polynomials ⓘ
computes	greatest common divisor of polynomials ⓘ
domain	polynomials over a Euclidean domain ⓘ polynomials over a field ⓘ polynomials over a principal ideal domain ⓘ
extendedVariantComputes	Bezout coefficients for polynomials ⓘ
extendedVariantUsedFor	Chinese remainder theorem for polynomials ⓘ linked to: Chinese remainder theorem in commutative algebra computing inverses modulo a polynomial ⓘ
fieldOfStudy	abstract algebra ⓘ algebra ⓘ computational algebra ⓘ
finalNonzeroRemainderIs	greatest common divisor ⓘ
hasStep	repeated polynomial division with remainder ⓘ replace pair by divisor and remainder ⓘ
hasVariant	extended Euclidean algorithm for polynomials ⓘ pseudo-division based Euclidean algorithm ⓘ subresultant polynomial remainder sequence algorithm ⓘ
historicalOrigin	generalization of Euclid’s algorithm to polynomial rings ⓘ
input	two polynomials ⓘ
normalization	gcd often chosen to be monic ⓘ
output	gcd up to multiplication by a unit ⓘ greatest common divisor polynomial ⓘ
property	gcd is unique up to multiplication by a nonzero constant ⓘ
relatedConcept	polynomial remainder sequence ⓘ principal ideal in a polynomial ring ⓘ resultant of polynomials ⓘ
reliesOn	Euclidean domain structure of coefficient ring ⓘ
requires	degree function on polynomials ⓘ
satisfies	any common divisor of a and b divides gcd(a,b) ⓘ gcd(a,b) divides both a and b ⓘ
terminatesWhen	remainder is zero ⓘ
usedIn	coding theory ⓘ computation of greatest common divisors in computer algebra systems ⓘ computation of resultants ⓘ construction of minimal polynomials ⓘ control theory ⓘ factorization of polynomials ⓘ signal processing ⓘ simplification of rational functions ⓘ
uses	polynomial long division ⓘ

How these facts were elicited