Abel–Ruffini theorem
E63710
The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Abel–Ruffini theorem canonical | 2 |
| Abel’s impossibility theorem | 1 |
| Mémoire sur les conditions de résolubilité des équations par radicaux | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T511390 — 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: Abel–Ruffini theorem Context triple: [Niels Henrik Abel, knownFor, Abel–Ruffini theorem]
-
A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
B.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
E.
Évariste Galois
Évariste Galois was a pioneering 19th-century French mathematician whose foundational work in group theory and the theory of equations gave rise to modern Galois theory.
- 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: Abel–Ruffini theorem Target entity description: The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
-
A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
B.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
E.
Évariste Galois
Évariste Galois was a pioneering 19th-century French mathematician whose foundational work in group theory and the theory of equations gave rise to modern Galois theory.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in algebra ⓘ |
| alsoKnownAs |
Abel–Ruffini theorem
ⓘ
surface form:
Abel’s impossibility theorem
|
| appliesTo |
general polynomial equations of degree n with n ≥ 5
ⓘ
general quintic equations ⓘ |
| assumes | coefficients in a field of characteristic zero, typically the rational numbers ⓘ |
| characterizes | solvability of polynomial equations by radicals via solvability of their Galois groups ⓘ |
| clarifies |
limits of algebraic methods based solely on radicals
ⓘ
that some specific quintic equations can be solvable by radicals even though the general quintic is not ⓘ |
| concerns |
equations of degree five or higher
ⓘ
polynomial equations ⓘ solvability by radicals ⓘ |
| consequence | no algorithm using only radicals and arithmetic operations can solve all quintic equations ⓘ |
| doesNotApplyTo |
cubic equations
ⓘ
quadratic equations ⓘ quartic equations ⓘ |
| field |
Galois theory
ⓘ
abstract algebra ⓘ algebra ⓘ group theory ⓘ |
| historicalDevelopment | Niels Henrik Abel gave the first complete proof of the impossibility of solving the general quintic by radicals ⓘ |
| historicalPrecursor | work of Paolo Ruffini on the unsolvability of the general quintic ⓘ |
| implies | there is no radical expression for the roots of a generic polynomial of degree five or higher ⓘ |
| importance | fundamental result in the theory of polynomial equations ⓘ |
| influenced |
development of modern algebra
ⓘ
theory of equations ⓘ |
| involves |
field extensions
ⓘ
radical extensions ⓘ symmetric group S_n ⓘ |
| keyObservation | for n ≥ 5 the symmetric group S_n is not solvable ⓘ |
| logicalType | impossibility theorem ⓘ |
| mathematicalDomain |
commutative algebra
ⓘ
field theory ⓘ |
| namedAfter |
Niels Henrik Abel
ⓘ
Paolo Ruffini ⓘ |
| relatedTo |
Galois theory
ⓘ
surface form:
Galois’ theory of equations
fundamental theorem of Galois theory ⓘ quintic equation ⓘ solvable by radicals ⓘ unsolvability results in mathematics ⓘ |
| states |
no formula using only radicals and field operations exists that expresses the roots of a general quintic in terms of its coefficients
ⓘ
there is no general solution in radicals for polynomial equations of degree at least five ⓘ |
| status | proven ⓘ |
| typicalFormulation | the general polynomial equation of degree five or higher with arbitrary coefficients is not solvable by radicals ⓘ |
| usesConcept |
Galois group
ⓘ
permutation group of the roots ⓘ solvable group ⓘ |
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: Abel–Ruffini theorem Description of subject: The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Abel’s impossibility theorem
subject surface form:
Évariste Galois
this entity surface form:
Mémoire sur les conditions de résolubilité des équations par radicaux