Editing Abel–Ruffini theorem (section)

{{short description|Equations of degree 5 or higher cannot be solved by radicals}}
{{Distinguish|Abel's theorem}}

In [[mathematics]], the '''Abel–Ruffini theorem''' (also known as '''Abel's impossibility theorem''') states that there is no [[solution in radicals]] to general [[algebraic equation|polynomial equations]] of [[quintic equation|degree five]] or higher with arbitrary [[coefficients]]. Here, ''general'' means that the coefficients of the equation are viewed and manipulated as [[indeterminate (variable)|indeterminates]].

The theorem is named after [[Paolo Ruffini (mathematician)|Paolo Ruffini]], who made an incomplete proof in 1799<ref name="Ayoub"/> (which was refined and completed in 1813<ref>{{Cite book|last=Ruffini|first=Paolo|url=https://books.google.com/books?id=sxW1QhMTtp8C&dq=Riflessioni+intorno+alla+soluzione+delle+equazioni+algebraiche+generali&pg=PA4|title=Riflessioni intorno alla soluzione delle equazioni algebraiche generali opuscolo del cav. dott. Paolo Ruffini ...|date=1813|publisher=presso la Societa Tipografica|language=it}}</ref> and accepted by [[Cauchy]]) and [[Niels Henrik Abel]], who provided a proof in 1824.<ref name="Abel1">{{Citation|last=Abel|first=Niels Henrik|author-link=Niels Henrik Abel|chapter=Mémoire sur les équations algébriques, ou l'on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré|chapter-url=http://www.abelprisen.no/nedlastning/verker/oeuvres_1881_del1/oeuvres_completes_de_abel_nouv_ed_1_kap03_opt.pdf|title=Œuvres Complètes de Niels Henrik Abel|volume=I|edition=2nd|editor-last=Sylow|editor-first=Ludwig|editor-link=Peter Ludwig Mejdell Sylow|editor2-last=Lie|editor2-first=Sophus|editor-link2=Sophus Lie|year=1881|orig-year=1824|pages=28–33|publisher=[[Grøndahl & Søn Forlag|Grøndahl & Søn]]|language=fr}}</ref><ref name="Abel2">{{Citation|last=Abel|first=Niels Henrik|author-link=Niels Henrik Abel|chapter=Démonstration de l'impossibilité de la résolution algébrique des équations générales qui passent le quatrième degré|title=Œuvres Complètes de Niels Henrik Abel|volume=I|edition=2nd|editor-last=Sylow|editor-first=Ludwig|editor-link=Peter Ludwig Mejdell Sylow|editor2-last=Lie|editor2-first=Sophus|editor-link2=Sophus Lie|year=1881|orig-year=1826|pages=66–87|chapter-url=http://www.abelprisen.no/nedlastning/verker/oeuvres_1839/oeuvres_completes_de_abel_1_kap02_opt.pdf|publisher=[[Grøndahl & Søn Forlag|Grøndahl & Søn]]|language=fr}}</ref>

''Abel–Ruffini theorem'' refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial. This improved statement follows directly from {{slink|Galois theory|A non-solvable quintic example}}. Galois theory implies also that 
:<math>x^5-x-1=0</math>
is the simplest equation that cannot be solved in radicals, and that ''[[almost all]]'' polynomials of degree five or higher cannot be solved in radicals.

The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the [[quadratic formula]], the [[cubic formula]], and the [[quartic formula]] for degrees two, three, and four, respectively.