Zassenhaus algorithm for factoring polynomials over the rationals
E827382
The Zassenhaus algorithm for factoring polynomials over the rationals is a classical computational method that reduces rational polynomial factorization to modular factorization and then recombines the results using lifting techniques.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Zassenhaus algorithm for factoring polynomials over the rationals canonical | 1 |
| Zassenhaus algorithm in computer algebra | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9867909 — 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: Zassenhaus algorithm for factoring polynomials over the rationals Context triple: [Hans Zassenhaus, notableWork, Zassenhaus algorithm for factoring polynomials over the rationals]
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A.
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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B.
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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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.
Lenstra elliptic-curve factorization method
The Lenstra elliptic-curve factorization method is an integer factorization algorithm that uses properties of elliptic curves over finite fields to efficiently find nontrivial factors of large numbers, especially those with relatively small prime divisors.
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E.
Euclidean algorithm for polynomials
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.
- 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: Zassenhaus algorithm for factoring polynomials over the rationals Target entity description: The Zassenhaus algorithm for factoring polynomials over the rationals is a classical computational method that reduces rational polynomial factorization to modular factorization and then recombines the results using lifting techniques.
-
A.
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.
-
B.
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.
-
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.
Lenstra elliptic-curve factorization method
The Lenstra elliptic-curve factorization method is an integer factorization algorithm that uses properties of elliptic curves over finite fields to efficiently find nontrivial factors of large numbers, especially those with relatively small prime divisors.
-
E.
Euclidean algorithm for polynomials
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
computational algebra algorithm
ⓘ
polynomial factorization algorithm ⓘ symbolic computation method ⓘ |
| appliesTo | univariate polynomials over the rationals ⓘ |
| assumes | polynomial is squarefree over the rationals or has been squarefree decomposed ⓘ |
| basedOn |
Hensel lifting
ⓘ
linked to:
Hensel’s lemma
modular factorization ⓘ |
| category | exact polynomial algorithms ⓘ |
| complexityDependsOn |
degree of the polynomial
ⓘ
size of coefficients ⓘ |
| contrastWith | probabilistic factorization algorithms over finite fields ⓘ |
| domain |
computational algebra
ⓘ
computer algebra systems ⓘ |
| goal | reduce rational polynomial factorization to modular factorization ⓘ |
| historicalPeriod | 20th century ⓘ |
| improvesOn | naive integer factor search for polynomial factorization ⓘ |
| input | polynomial with rational coefficients ⓘ |
| involves |
content-primitive factorization of integer polynomials
ⓘ
squarefree factorization as a preprocessing step ⓘ |
| mathematicalField |
algebra
ⓘ
algorithmic algebra ⓘ number theory ⓘ |
| namedAfter | Hans Zassenhaus ⓘ |
| output | factorization into irreducible polynomials over the rationals ⓘ |
| property |
deterministic
ⓘ
exact arithmetic algorithm ⓘ |
| relatedTo |
Berlekamp algorithm
ⓘ
Cantor–Zassenhaus algorithm ⓘ Kronecker factorization method ⓘ |
| requires | bound on coefficients of factors for recombination step ⓘ |
| step |
choose a suitable prime not dividing the content or discriminant
ⓘ
clear denominators to obtain an integer polynomial ⓘ factor the polynomial modulo the chosen prime ⓘ lift modular factors to higher powers of the prime using Hensel lifting ⓘ recombine lifted factors to obtain rational factors ⓘ |
| typicalImplementationLanguage | computer algebra system kernels such as Maple, Mathematica, or SageMath ⓘ |
| usedFor |
computing algebraic number fields defined by polynomial equations
ⓘ
finding irreducible decomposition of rational polynomials ⓘ simplifying algebraic expressions ⓘ |
| usedIn |
implementation of factor() in computer algebra systems
ⓘ
symbolic integration and simplification routines ⓘ |
| uses |
Chinese remainder techniques
ⓘ
combinatorial search over lifted factors ⓘ reduction modulo a prime ⓘ |
| worksOver |
field of rational numbers
ⓘ
ring of integers via content-primitive decomposition ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Zassenhaus algorithm for factoring polynomials over the rationals Description of subject: The Zassenhaus algorithm for factoring polynomials over the rationals is a classical computational method that reduces rational polynomial factorization to modular factorization and then recombines the results using lifting techniques.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Zassenhaus algorithm for factoring polynomials over the rationals