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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Observed surface forms (1)

Surface form	Occurrences
Abel’s impossibility theorem	1

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 ⓘ

Referenced by (3)

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

Abel–Ruffini theorem → alsoKnownAs → Abel–Ruffini theorem ⓘ

this entity surface form: Abel’s impossibility theorem

Niels Henrik Abel → knownFor → Abel–Ruffini theorem ⓘ

Niels → knownFor → Abel–Ruffini theorem ⓘ

subject surface form: Niels Henrik Abel