Editing Galois theory (section)

== History ==
{{see also|Abstract algebra#Early group theory}}

=== Pre-history ===
Galois' theory originated in the study of [[symmetric function]]s – the coefficients of a [[monic polynomial]] are ([[up to]] sign) the [[elementary symmetric polynomial]]s in the roots. For instance, {{math|1=(''x'' – ''a'')(''x'' – ''b'') = ''x''<sup>2</sup> – (''a'' + ''b'')''x'' + ''ab''}}, where 1, {{math|''a'' + ''b''}} and {{math|''ab''}} are the elementary polynomials of degree 0, 1 and 2 in two variables.

This was first formalized by the 16th-century French mathematician [[François Viète]], in [[Viète's formulas]], for the case of positive real roots. In the opinion of the 18th-century British mathematician [[Charles Hutton]],<ref>{{Harvnb|Funkhouser|1930}}</ref> the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th-century French mathematician [[Albert Girard]]; Hutton writes:
<blockquote>...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.</blockquote>

In this vein, the [[discriminant]] is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots. See {{slink|Discriminant#Low_degrees}} for details.

The cubic was first partly solved by the 15–16th-century Italian mathematician [[Scipione del Ferro]], who did not however publish his results; this method, though, only solved one type of cubic equation. This solution was then rediscovered independently in 1535 by [[Niccolò Fontana Tartaglia]], who shared it with [[Gerolamo Cardano]], asking him to not publish it. Cardano then extended this to numerous other cases, using similar arguments; see more details at [[Cubic equation#Cardano's method|Cardano's method]]. After the discovery of del Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his solution in his 1545 ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]].''<ref>{{harvnb|Cardano|1545}}</ref> His student [[Lodovico Ferrari]] solved the quartic polynomial; his solution was also included in ''Ars Magna.'' In this book, however, Cardano did not provide a "general formula" for the solution of a cubic equation, as he had neither [[complex numbers]] at his disposal, nor the algebraic notation to be able to describe a general cubic equation. With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case, but Cardano did not know this.  It was [[Rafael Bombelli]] who managed to understand how to work with complex numbers in order to solve all forms of cubic equation.

A further step was the 1770 paper ''Réflexions sur la résolution algébrique des équations'' by the French-Italian mathematician [[Joseph Louis Lagrange]], in his method of [[Lagrange resolvents]], where he analyzed Cardano's and Ferrari's solution of cubics and quartics by considering them in terms of ''permutations'' of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois' theory. Crucially, however, he did not consider ''composition'' of permutations. Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree.

The quintic was almost proven to have no general solutions by radicals by [[Paolo Ruffini (mathematician)|Paolo Ruffini]] in 1799, whose key insight was to use [[permutation group|permutation ''groups'']], not just a single permutation. His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of the Norwegian mathematician [[Niels Henrik Abel]], who published a proof in 1824, thus establishing the [[Abel–Ruffini theorem]].

While Ruffini and Abel established that the ''general'' quintic could not be solved, some ''particular'' quintics can be solved, such as {{math|''x''<sup>5</sup> - 1 {{=}} 0}}, and the precise criterion by which a ''given'' quintic or higher polynomial could be determined to be solvable or not was given by [[Évariste Galois]], who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its [[Galois group]] – had a certain structure – in modern terms, whether or not it was a [[solvable group]]. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degrees.{{Citation needed|date=January 2025}}

=== Galois' writings ===
[[File:Evariste Galois.jpg|alt=Évariste Galois|thumb|A portrait of Évariste Galois aged about 15]]
In 1830 Galois (at the age of 18) submitted to the [[Paris Academy of Sciences]] a memoir on his theory of solvability by radicals; Galois' paper 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 then died in a duel in 1832, and his paper, "''Mémoire sur les conditions de résolubilité des équations par radicaux''", remained unpublished until 1846 when it was published by [[Joseph Liouville]] accompanied by some of his own explanations.<ref name="Tignol2001">{{cite book|first=Jean-Pierre|last=Tignol| author-link=Jean-Pierre Tignol |title=Galois' Theory of Algebraic Equations|url=https://archive.org/details/galoistheoryalge00tign_325|url-access=limited|year=2001|publisher=World Scientific|isbn=978-981-02-4541-2|pages=[https://archive.org/details/galoistheoryalge00tign_325/page/n242 232]–3, 302}}</ref> Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843.<ref>Stewart, 3rd ed., p. xxiii</ref> According to Allan Clark, Galois's characterization "dramatically supersedes the work of Abel and Ruffini."<ref name="Clark1984">{{cite book|first=Allan|last=Clark|title=Elements of Abstract Algebra|year=1984|orig-year=1971|publisher=Courier |isbn=978-0-486-14035-3|page=131}}</ref>

=== Aftermath ===
Galois' theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method.<ref name="Wussing2007">{{cite book|first=Hans|last=Wussing|title=The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory|year=2007|publisher=Courier |isbn=978-0-486-45868-7|page=118}}</ref> [[Joseph Alfred Serret]] who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook ''Cours d'algèbre supérieure''. Serret's pupil, [[Camille Jordan]], had an even better understanding reflected in his 1870 book ''Traité des substitutions et des équations algébriques''. Outside France, Galois' theory remained more obscure for a longer period. In Britain, [[Arthur Cayley|Cayley]] failed to grasp its depth and popular British algebra textbooks did not even mention Galois' theory until well after the turn of the century. In Germany, Kronecker's writings focused more on Abel's result. [[Richard Dedekind|Dedekind]] wrote little about Galois' theory, but lectured on it at Göttingen in 1858, showing a very good understanding.<ref name="Scharlau1981">{{cite book |first1=Winfried |last1=Scharlau |first2=Ilse |last2=Dedekind |first3=Richard |last3=Dedekind |title=Richard Dedekind 1831–1981; eine Würdigung zu seinem 150. Geburtstag |publisher=Vieweg |location=Braunschweig |year=1981 |isbn=9783528084981 |url=http://www.gbv.de/dms/ilmenau/toc/023900512.PDF}}</ref> [[Eugen Netto]]'s books of the 1880s, based on Jordan's ''Traité'', made Galois theory accessible to a wider German and American audience as did [[Heinrich Martin Weber]]'s  1895 algebra textbook.<ref name="GaloisNeumann2011">{{cite book|first1=Évariste|last1=Galois|first2=Peter M.|last2=Neumann|title=The Mathematical Writings of Évariste Galois|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-104-0|page=10}}</ref>