Editing Évariste Galois (section)

== Contributions to mathematics ==
[[File:E. Galois Letter.jpg|thumb|right|The final page of Galois's mathematical testament, in his own hand. The phrase "to decipher all this mess" ("déchiffrer tout ce gâchis") is on the second to the last line.]]

From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated 29 May 1832, two days before Galois's death:<ref name="chevalier-letter" />

{{blockquote|''Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l'importance des théorèmes.''

''Après cela, il y aura, j'espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.''

(Ask [[Carl Gustav Jacob Jacobi|Jacobi]] or [[Carl Friedrich Gauss|Gauss]] publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)}}

Within the 60 or so pages of Galois's collected works are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.<ref name="lie">{{cite conference
 | last = Lie
 | first = Sophus
 | year = 1895
 | title = Influence de Galois sur le Développement des Mathématiques
 | book-title = Le centenaire de l'École Normale 1795–1895
 | publisher = Hachette
}}</ref><ref>See also: Sophus Lie, [http://gallica.bnf.fr/ark:/12148/bpt6k290623/f71.image "Influence de Galois sur le développement des mathématiques"] in: Évariste Galois, ''Oeuvres Mathématiques publiées en 1846 dans le'' Journal de Liouville (Sceaux, France: Éditions Jacques Gabay, 1989), appendix pages 1–9.</ref> 
His work has been compared to that of [[Niels Henrik Abel]] (1802–1829), a contemporary mathematician who also died at a very young age, and much of their work had significant overlap.

=== Algebra ===
While many mathematicians before Galois gave consideration to what are now known as [[group (algebra)|groups]], he was the first one to use the word ''group'' (in French ''groupe'') in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as [[group theory]]. He called the decomposition of a group into its left and right [[coset]]s a ''proper decomposition'' if the left and right cosets coincide, which leads to the notion of what today are known as [[normal subgroup]]s.<ref name="chevalier-letter" /> He also introduced the concept of a [[finite field]] (also known as a [[Galois field]] in his honor) in essentially the same form as it is understood today.<ref name="numtheory" />

In his last letter to Chevalier<ref name="chevalier-letter" /> and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:
*He constructed the [[General linear group#Over finite fields|general linear group over a prime field]], GL(''ν'', ''p'') and computed its order, in studying the Galois group of the general equation of degree ''p<sup>ν</sup>''.<ref>Letter, p. 410</ref>
*He constructed the [[projective special linear group]] PSL(2,''p''). Galois constructed them as fractional linear transforms, and observed that they were simple except if ''p'' was 2 or 3.<ref>Letter, p. 411</ref> These were the second family of finite [[simple group]]s, after the [[alternating group]]s.<ref name="raw">{{cite book | last1=Wilson | first1=Robert A. | author-link = Robert Arnott Wilson | title = The finite simple groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]]|volume= 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | zbl=1203.20012 | year=2009 |chapter = Chapter 1: Introduction |chapter-url=http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps}}</ref>
*He noted the [[exceptional object|exceptional fact]] that PSL(2,''p'') is simple and [[Projective linear group#Action on p points|acts on ''p'' points]] if and only if ''p'' is 5, 7, or 11.<ref>Letter, pp. 411–412</ref><ref>{{cite web| url = http://people.math.umass.edu/~tevelev/475_2016/galois_lc.pdf| title = Galois's last letter, translated}}</ref>

=== Galois theory ===
{{Main|Galois theory}}
Galois's most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a [[polynomial]] equation is related to the structure of a group of [[permutation]]s associated with the roots of the polynomial, the [[Galois group]] of the polynomial. He found that an equation could be solved in [[nth root|radicals]] if one can find a series of subgroups of its Galois group, each one normal in its successor with [[abelian group|abelian]] quotient, that is, its Galois group is [[solvable group|solvable]]. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the [[theory of equations]] to which Galois originally applied it.<ref name="lie" />

=== Analysis ===
Galois also made some contributions to the theory of [[Abelian integral]]s and [[continued fraction]]s.

As written in his last letter,<ref name="chevalier-letter" /> Galois passed from the study of elliptic functions to consideration of the integrals of the most general algebraic differentials, today called Abelian integrals. He classified these integrals into three categories.

=== Continued fractions ===
In his first paper in 1828,<ref name="continuedfraction" /> Galois proved that the regular continued fraction which represents a quadratic [[Nth root|surd]] ''ζ'' is purely periodic if and only if ''ζ'' is a [[Periodic continued fraction#Reduced surds|reduced surd]], that is, <math>\zeta > 1</math> and its [[Conjugate (algebra)|conjugate]] <math>\eta</math>
satisfies <math>-1 < \eta < 0</math>.

In fact, Galois showed more than this. He also proved that if ''ζ'' is a reduced quadratic surd and ''η'' is its conjugate, then the continued fractions for ''ζ'' and for (−1/''η'') are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have

:<math>
\begin{align}
\zeta& = [\,\overline{a_0;a_1,a_2,\dots,a_{m-1}}\,]\\[3pt]
\frac{-1}{\eta}& = [\,\overline{a_{m-1};a_{m-2},a_{m-3},\dots,a_0}\,]\,
\end{align}
</math>

where ''ζ'' is any reduced quadratic surd, and ''η'' is its conjugate.

From these two theorems of Galois a result already known to Lagrange can be deduced. If ''r''&nbsp;>&nbsp;1 is a rational number that is not a perfect square, then

:<math>
\sqrt{r} = \left[\,a_0;\overline{a_1,a_2,\dots,a_2,a_1,2a_0}\,\right].
</math>

In particular, if ''n'' is any non-square positive integer, the regular continued fraction expansion of √''n'' contains a repeating block of length ''m'', in which the first ''m''&nbsp;−&nbsp;1 partial denominators form a [[palindrome|palindromic]] string.