Euclidean algorithm for polynomials
E646780
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Euclidean algorithm | 1 |
| Euclidean algorithm for polynomials canonical | 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]
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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.
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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.
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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.
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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.
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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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Euclidean algorithm for polynomials Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Euclidean algorithm