Abel–Ruffini theorem

E63710

mathematical theorem theorem in algebra

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.

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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’s impossibility theorem ⓘ linked to: Abel–Ruffini 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 of equations ⓘ linked to: Galois theory 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

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Niels Henrik Abel → knownFor → Abel–Ruffini theorem ⓘ

subject linked to: Niels

Abel–Ruffini theorem → alsoKnownAs → Abel’s impossibility theorem ⓘ

linked to: Abel–Ruffini theorem

Évariste Galois → notableWork → Mémoire sur les conditions de résolubilité des équations par radicaux ⓘ

subject linked to: Galois