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Abel–Ruffini theorem
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==History== Around 1770, [[Joseph Louis Lagrange]] began the groundwork that unified the many different methods that had been used up to that point to solve equations, relating them to the theory of groups of [[permutation]]s, in the form of [[Lagrange resolvents]].<ref>{{Citation|last=Lagrange|first=Joseph-Louis|author-link=Joseph Louis Lagrange|chapter=Réflexions sur la résolution algébrique des équations|title=Œuvres de Lagrange|editor-last=Serret|editor-first=Joseph-Alfred|editor-link=Joseph Alfred Serret|year=1869|orig-year=1771|publisher=Gauthier-Villars|pages=205–421|volume=III}}</ref> This innovative work by Lagrange was a precursor to Galois theory, and its failure to develop solutions for equations of fifth and higher degrees hinted that such solutions might be impossible, but it did not provide conclusive proof. The first person who conjectured that the problem of solving quintics by radicals might be impossible to solve was [[Carl Friedrich Gauss]], who wrote in 1798 in section 359 of his book ''[[Disquisitiones Arithmeticae]]'' (which would be published only in 1801) that "there is little doubt that this problem does not so much defy modern methods of analysis as that it proposes the impossible". The next year, in his [[thesis]], he wrote "After the labors of many geometers left little hope of ever arriving at the resolution of the general equation algebraically, it appears more and more likely that this resolution is impossible and contradictory." And he added "Perhaps it will not be so difficult to prove, with all rigor, the impossibility for the fifth degree. I shall set forth my investigations of this at greater length in another place." Actually, Gauss published nothing else on this subject.<ref name="Ayoub"/> [[File:Ruffini - Teoria generale delle equazioni, 1799 - 1366896.jpg|thumb|[[Paolo Ruffini]], ''Teoria generale delle equazioni'', 1799]] The theorem was first nearly proved by [[Paolo Ruffini (mathematician)|Paolo Ruffini]] in 1799.<ref>{{Citation|last=Ruffini|first=Paolo|author-link=Paolo Ruffini|year=1799|title=Teoria generale delle equazioni, in cui si dimostra impossibile la soluzione algebraica delle equazioni generali di grado superiore al quarto|publisher=Stamperia di S. Tommaso d'Aquino|language=it}}</ref> He sent his proof to several mathematicians to get it acknowledged, amongst them Lagrange (who did not reply) and [[Augustin-Louis Cauchy]], who sent him a letter saying: "Your memoir on the general solution of equations is a work which I have always believed should be kept in mind by mathematicians and which, in my opinion, proves conclusively the algebraic unsolvability of general equations of higher than fourth degree."<ref name="Kiernan">{{Citation|last=Kiernan|first=B. Melvin|journal=[[Archive for History of Exact Sciences]]|title=The Development of Galois Theory from Lagrange to Artin|volume=8|issue=1/2|pages=40–154|jstor=41133337|doi=10.1007/BF00327219|year=1971|s2cid=121442989|mr=1554154}}</ref> However, in general, Ruffini's proof was not considered convincing. Abel wrote: "The first and, if I am not mistaken, the only one who, before me, has sought to prove the impossibility of the algebraic solution of general equations is the mathematician Ruffini. But his memoir is so complicated that it is very difficult to determine the validity of his argument. It seems to me that his argument is not completely satisfying."<ref name="Kiernan"/><ref>{{Citation|last=Abel|first=Niels Henrik|author-link=Niels Henrik Abel|chapter=Sur la resolution algébrique des équations|chapter-url=http://www.abelprize.no/nedlastning/verker/oeuvres_1881_del2/oeuvres_completes_de_abel_nouv_ed_2_kap18_opt.pdf|title=Œuvres Complètes de Niels Henrik Abel|volume=II|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=1828|pages=217–243|publisher=[[Grøndahl & Søn Forlag|Grøndahl & Søn]]|language=fr}}</ref> The proof also, as it was discovered later, was incomplete. Ruffini assumed that all radicals that he was dealing with could be expressed from the roots of the polynomial using field operations alone; in modern terms, he assumed that the radicals belonged to the splitting field of the polynomial. To see why this is really an extra assumption, consider, for instance, the polynomial <math>P(x)=x^{3}-15x-20</math>. According to [[cubic equation#Cardano's formula|Cardano's formula]], one of its roots (all of them, actually) can be expressed as the sum of a cube root of <math>10+5i</math> with a cube root of <math>10-5i</math>. On the other hand, since <math>P(-3)<0</math>, <math>P(-2)>0</math>, <math>P(-1)<0</math>, and <math>P(5)>0</math>, the roots <math>r_1</math>, <math>r_2</math>, and <math>r_3</math> of <math>P(x)</math> are all real and therefore the field <math>\mathbf{Q}(r_1,r_2,r_3)</math> is a subfield of <math>\mathbf{R}</math>. But then the numbers <math>10 \pm 5i</math> cannot belong to <math>\mathbf{Q}(r_1,r_2,r_3)</math>. While Cauchy either did not notice Ruffini's assumption or felt that it was a minor one, most historians believe that the proof was not complete until Abel proved the theorem on natural irrationalities, which asserts that the assumption holds in the case of general polynomials.<ref name="Tignol_Abel"/><ref>{{Citation|last=Stewart|first=Ian| author-link=Ian Stewart (mathematician)|title=Galois Theory|chapter=The Idea Behind Galois Theory|publisher=[[CRC Press]]|isbn=978-1-4822-4582-0|year=2015|edition=4th}}</ref> The Abel–Ruffini theorem is thus generally credited to Abel, who published a proof compressed into just six pages in 1824.<ref name="Abel1"/> (Abel adopted a very terse style to save paper and money: the proof was printed at his own expense.<ref name= "Pesic" />) A more elaborated version of the proof would be published in 1826.<ref name="Abel2"/> Proving that the general quintic (and higher) equations were unsolvable by radicals did not completely settle the matter, because the Abel–Ruffini theorem does not provide necessary and sufficient conditions for saying precisely which quintic (and higher) equations are unsolvable by radicals. Abel was working on a complete characterization when he died in 1829.<ref name="Tignol2001"/> According to [[Nathan Jacobson]], "The proofs of Ruffini and of Abel [...] were soon superseded by the crowning achievement of this line of research: Galois' discoveries in the theory of equations."<ref name="Jacobson">{{Citation |last=Jacobson |first=Nathan |author-link=Nathan Jacobson |year=2009 |title=Basic Algebra |edition=2nd |volume=1 |publisher=Dover |isbn=978-0-486-47189-1|chapter=Galois Theory of Equations}}</ref> In 1830, Galois (at the age of 18) submitted to the [[Paris Academy of Sciences]] a memoir on his theory of solvability by radicals, which was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois was aware of the contributions of Ruffini and Abel, since he wrote "It is a common truth, today, that the general equation of degree greater than {{math|4}} cannot be solved by radicals... this truth has become common (by hearsay) despite the fact that geometers have ignored the proofs of Abel and Ruffini..."<ref name="Ayoub">{{Citation|last=Ayoub|first=Raymond G.|title=Paolo Ruffini's Contributions to the Quintic|journal=[[Archive for History of Exact Sciences]]|doi=10.1007/BF00357046|volume=22|issue=3|pages=253–277|year=1980|jstor=41133596|mr=606270|zbl=0471.01008|s2cid=123447349}}</ref> Galois then died in 1832 and his paper ''Mémoire sur les conditions de resolubilité des équations par radicaux''<ref>{{Citation|last=Galois|first=Évariste|author-link=Évariste Galois|journal=[[Journal de Mathématiques Pures et Appliquées]]|title=Mémoire sur les conditions de resolubilité des équations par radicaux|pages=417–433|url=https://www.bibnum.education.fr/sites/default/files/galois_memoire_sur_la_resolubiblite.pdf|year=1846|volume=XI|language=fr}}</ref> remained unpublished until 1846, when it was published by [[Joseph Liouville]] accompanied by some of his own explanations.<ref name="Tignol2001">{{Citation|last=Tignol|first=Jean-Pierre|title=Galois' Theory of Algebraic Equations|edition=2nd| publisher=[[World Scientific]]|year=2016|isbn=978-981-4704-69-4|zbl=1333.12001|chapter=Galois}}</ref> Prior to this publication, Liouville announced Galois' result to the academy in a speech he gave on 4 July 1843.<ref name="Stewart"/> A simplification of Abel's proof was published by [[Pierre Wantzel]] in 1845.<ref name="Wantzel">{{Citation|last=Wantzel|first=Pierre| title=Démonstration de l'impossibilité de résoudre toutes les équations algébriques avec des radicaux| url=http://www.numdam.org/item?id=NAM_1845_1_4__57_0|year=1845|author-link=Pierre Wantzel|journal=[[Nouvelles Annales de Mathématiques]]|volume=4|pages=57–65|language=fr}}</ref> When Wantzel published it, he was already aware of the contributions by Galois and he mentions that, whereas Abel's proof is valid only for general polynomials, Galois' approach can be used to provide a concrete polynomial of degree 5 whose roots cannot be expressed in radicals from its coefficients. In 1963, [[Vladimir Arnold]] discovered a [[topology|topological]] proof of the Abel–Ruffini theorem,<ref>{{Citation|last=Alekseev|first=Valeriy B.|title=Abel's Theorem in Problems and Solutions: Based on the Lectures of Professor V. I. Arnold|publisher=[[Springer Science+Business Media|Kluwer Academic Publishers]]|year=2004|zbl=1065.12001|isbn=1-4020-2186-0|mr=2110624}}</ref><ref>{{Citation|last=Goldmakher|first=Leo|title=Arnold's Elementary Proof of the Insolvability of the Quintic| url=http://web.williams.edu/Mathematics/lg5/ArnoldQuintic.pdf}}</ref> which served as a starting point for [[topological Galois theory]].<ref>{{Citation|last=Khovanskii|first=Askold|author-link=Askold Khovanskii|title=Topological Galois Theory: Solvability and Unsolvability of Equations in Finite Terms|publisher=[[Springer Science+Business Media|Springer-Verlag]]|year=2014|isbn=978-3-642-38870-5|doi=10.1007/978-3-642-38871-2|series=Springer Monographs in Mathematics}}</ref>
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