Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and political activist. While still in his teens, he was able to determine a

necessary and sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

for a

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

to be solvable by radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for

Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

and

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, two major branches of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

. Galois was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830. As a result of his political activism, he was arrested repeatedly, serving one jail sentence of several months. For reasons that remain obscure, shortly after his release from prison, Galois fought in a

duel A duel is an arranged engagement in combat between two people with matched weapons. During the 17th and 18th centuries (and earlier), duels were mostly single combats fought with swords (the rapier and later the small sword), but beginning in ...

and died of the wounds he suffered.

Life

Early life

Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (née Demante). His father was a Republican and was head of Bourg-la-Reine's

liberal party The Liberal Party is any of many political parties around the world. The meaning of ''liberal'' varies around the world, ranging from liberal conservatism on the right to social liberalism on the left. For example, while the political systems ...

. His father became mayor of the village after

Louis XVIII Louis XVIII (Louis Stanislas Xavier; 17 November 1755 – 16 September 1824), known as the Desired (), was King of France from 1814 to 1824, except for a brief interruption during the Hundred Days in 1815. Before his reign, he spent 23 y ...

returned to the throne in 1814. His mother, the daughter of a

jurist A jurist is a person with expert knowledge of law; someone who analyzes and comments on law. This person is usually a specialist legal scholar, mostly (but not always) with a formal education in law (a law degree) and often a Lawyer, legal prac ...

, was a fluent reader of

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

and

classical literature Classics, also classical studies or Ancient Greek and Roman studies, is the study of classical antiquity. In the Western world, ''classics'' traditionally refers to the study of Ancient Greek and Roman literature and their original languages, ...

and was responsible for her son's education for his first twelve years. Louis-le-Grand--cour-honneur

In October 1823, he entered the

Lycée Louis-le-Grand The Lycée Louis-le-Grand (), also referred to simply as Louis-le-Grand or by its acronym LLG, is a public Lycée (French secondary school, also known as sixth form college) located on Rue Saint-Jacques (Paris), rue Saint-Jacques in central Par ...

where his teacher Louis Paul Émile Richard recognized his brilliance. At the age of 14, he began to take a serious interest in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. Galois found a copy of

Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transforma ...

's ''

Éléments de Géométrie ''Éléments'' () is a French bi-monthly magazine launched in September 1973 and associated with the Nouvelle Droite. It is published by the white nationalist thinktank GRECE. History Initially serving as the internal bulletin of GRECE, an ethn ...

'', which, it is said, he read "like a novel" and mastered at the first reading. At 15, he was reading the original papers of

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRéflexions sur la résolution algébrique des équations'' which likely motivated his later work on equation theory, and ''Leçons sur le calcul des fonctions'', work intended for professional mathematicians, yet his classwork remained uninspired and his teachers accused him of putting on the airs of a genius.

Budding mathematician

In 1828, Galois attempted the entrance examination for the

École Polytechnique (, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris. The school was founded in 1794 by mat ...

, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination. In that same year, he entered the École Normale (then known as l'École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him. Augustin Cauchy

In the following year Galois's first paper, on

simple continued fraction A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...

s, was published. It was at around the same time that he began making fundamental discoveries in the theory of

polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For example, x^5-3x+1=0 is a ...

s. He submitted two papers on this topic to the Academy of Sciences.

Augustin-Louis Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...

refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it is widely held that Cauchy recognized the importance of Galois's work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the academy's Grand Prize in Mathematics. Cauchy, an eminent mathematician of the time though with political views that were diametrically opposed to those of Galois, considered Galois's work to be a likely winner. On 28 July 1829, Galois's father died by suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt to enter the Polytechnique and failed yet again. It is undisputed that Galois was more than qualified; accounts differ on why he failed. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, which enraged Galois. The recent death of his father may have also influenced his behavior. Having been denied admission to the

École polytechnique (, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris. The school was founded in 1794 by mat ...

, Galois took the Baccalaureate examinations in order to enter the École normale. He passed, receiving his degree on 29 December 1829. His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research." Galois submitted his memoir on equation theory several times, but it was never published in his lifetime. Though his first attempt was refused by Cauchy, in February 1830 following Cauchy's suggestion he submitted it to the academy's secretary

Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analys ...

, to be considered for the Grand Prix of the academy. Unfortunately, Fourier died soon after, and the memoir was lost. The prize would be awarded that year to

Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...

posthumously and also to Carl Gustav Jacob Jacobi. Despite the lost memoir, Galois published three papers that year. One laid the foundations for

. The second was about the numerical resolution of equations (

root finding In numerical analysis, a root-finding algorithm is an algorithm for finding Zero of a function, zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be com ...

in modern terminology). The third was an important one in

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, in which the concept of a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

was first articulated.

Political firebrand

Galois lived during a time of political turmoil in France.

Charles X Charles X may refer to: * Charles X of France (1757–1836) * Charles X Gustav (1622–1660), King of Sweden * Charles, Cardinal de Bourbon (1523–1590), recognized as Charles X of France but renounced the royal title See also * * King Charle ...

had succeeded

in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority. Charles, faced with political opposition from the chambers, staged a coup d'état, and issued his notorious July Ordinances, touching off the

July Revolution The French Revolution of 1830, also known as the July Revolution (), Second French Revolution, or ("Three Glorious ays), was a second French Revolution after French Revolution, the first of 1789–99. It led to the overthrow of King Cha ...

which ended with Louis Philippe becoming king. While their counterparts at the Polytechnique were making history in the streets, Galois, at the École Normale, was locked in by the school's director. Galois was incensed and wrote a blistering letter criticizing the director, which he submitted to the ''Gazette des Écoles'', signing the letter with his full name. Although the ''Gazette''s editor omitted the signature for publication, Galois was expelled. Although his expulsion would have formally taken effect on 4 January 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the

National Guard National guard is the name used by a wide variety of current and historical uniformed organizations in different countries. The original National Guard was formed during the French Revolution around a cadre of defectors from the French Guards. ...

. He divided his time between his mathematical work and his political affiliations. Due to controversy surrounding the unit, soon after Galois became a member, on 31 December 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois's former unit were arrested and charged with conspiracy to overthrow the government. In April 1831, the officers were acquitted of all charges, and on 9 May 1831, a banquet was held in their honor, with many illustrious people present, such as

Alexandre Dumas Alexandre Dumas (born Alexandre Dumas Davy de la Pailleterie, 24 July 1802 – 5 December 1870), also known as Alexandre Dumas , was a French novelist and playwright. His works have been translated into many languages and he is one of the mos ...

. The proceedings grew riotous. At some point, Galois stood and proposed a toast in which he said, "To Louis Philippe," with a

dagger A dagger is a fighting knife with a very sharp point and usually one or two sharp edges, typically designed or capable of being used as a cutting or stabbing, thrusting weapon.State v. Martin, 633 S.W.2d 80 (Mo. 1982): This is the dictionary or ...

above his cup. The republicans at the banquet interpreted Galois's toast as a threat against the king's life and cheered. He was arrested the following day at his mother's house and held in detention at Sainte-Pélagie prison until 15 June 1831, when he had his trial. Galois's defense lawyer cleverly claimed that Galois actually said, "To Louis-Philippe, ''if he betrays''," but that the qualifier was drowned out in the cheers. The prosecutor asked a few more questions, and perhaps influenced by Galois's youth, the jury acquitted him that same day. On the following

Bastille Day Bastille Day is the common name given in English-speaking countries to the national day of France, which is celebrated on 14 July each year. It is referred to, both legally and commonly, as () in French, though ''la fête nationale'' is also u ...

(14 July 1831), Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a loaded rifle, and a dagger. He was again arrested. During his stay in prison, Galois at one point drank alcohol for the first time at the goading of his fellow inmates. One of these inmates, François-Vincent Raspail, recorded what Galois said while drunk in a letter from 25 July. Excerpted from the letter: Raspail continues that Galois, still in a delirium, attempted suicide, and that he would have succeeded if his fellow inmates had not forcibly stopped him. Months later, when Galois's trial occurred on 23 October, he was sentenced to six months in prison for illegally wearing a uniform. While in prison, he continued to develop his mathematical ideas. He was released on 29 April 1832.

Final days

Galois returned to mathematics after his expulsion from the École Normale, although he continued to spend time in political activities. After his expulsion became official in January 1831, he attempted to start a private class in advanced algebra which attracted some interest, but this waned, as it seemed that his political activism had priority.

Siméon Denis Poisson Baron Siméon Denis Poisson (, ; ; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity ...

asked him to submit his work on the

theory of equations In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equation (mathematics), equations defined by a polynomial. The main problem of the theory of equations was to know when an al ...

, which he did on 17 January 1831. Around 4 July 1831, Poisson declared Galois's work "incomprehensible", declaring that " alois'sargument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor"; however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion." While Poisson's report was made before Galois's 14 July arrest, it took until October to reach Galois in prison. It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and decided to abandon publishing his papers through the academy and instead publish them privately through his friend Auguste Chevalier. Apparently, however, Galois did not ignore Poisson's advice, as he began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832, after which he was somehow talked into a duel. Galois's fatal duel took place on 30 May. The true motives behind the duel are obscure. There has been much speculation about them. What is known is that, five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair. Some archival investigation on the original letters suggests that the woman of romantic interest was Stéphanie-Félicie Poterin du Motel, the daughter of the physician at the hostel where Galois stayed during the last months of his life. Fragments of letters from her, copied by Galois himself (with many portions, such as her name, either obliterated or deliberately omitted), are available. The letters hint that Poterin du Motel had confided some of her troubles to Galois, and this might have prompted him to provoke the duel himself on her behalf. This conjecture is also supported by other letters Galois later wrote to his friends the night before he died. Galois's cousin, Gabriel Demante, when asked if he knew the cause of the duel, mentioned that Galois "found himself in the presence of a supposed uncle and a supposed fiancé, each of whom provoked the duel." Galois himself exclaimed: "I am the victim of an infamous coquette and her two dupes." As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville, who was actually one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois's first arrest. However, Dumas is alone in this assertion, and if he were correct it is unclear why d'Herbinville would have been involved. It has been speculated that he was Poterin du Motel's "supposed fiancé" at the time (she ultimately married someone else), but no clear evidence has been found supporting this conjecture. On the other hand, extant newspaper clippings from only a few days after the duel give a description of his opponent (identified by the initials "L.D.") that appear to more accurately apply to one of Galois's Republican friends, most probably Ernest Duchatelet, who was imprisoned with Galois on the same charges. Given the conflicting information available, the true identity of his killer may well be lost to history. Whatever the reasons behind the duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament, the famous letter to Auguste Chevalier outlining his ideas, and three attached manuscripts. Mathematician

Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." However, the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. In these final papers, he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the academy and other papers. Tombe galois

Early in the morning of 30 May 1832, he was shot in the

abdomen The abdomen (colloquially called the gut, belly, tummy, midriff, tucky, or stomach) is the front part of the torso between the thorax (chest) and pelvis in humans and in other vertebrates. The area occupied by the abdomen is called the abdominal ...

, was abandoned by his opponents and his own seconds, and was found by a passing farmer. He died the following morning at ten o'clock in the

Hôpital Cochin The Hôpital Cochin () is a hospital of public assistance in the rue du Faubourg-Saint-Jacques Paris 14e. It houses the central burn treatment centre of the city. The Hôpital Cochin is a section of the Faculté de Médecine Paris-Cité. It commem ...

(probably of

peritonitis Peritonitis is inflammation of the localized or generalized peritoneum, the lining of the inner wall of the abdomen and covering of the abdominal organs. Symptoms may include severe pain, swelling of the abdomen, fever, or weight loss. One pa ...

), after refusing the offices of a priest. His funeral ended in riots. There were plans to initiate an uprising during his funeral, but during the same time the leaders heard of General Jean Maximilien Lamarque's death and the rising was postponed without any uprising occurring until 5 June. Only Galois's younger brother was notified of the events prior to Galois's death. Galois was 20 years old. His

last words Last words are the final utterances before death. The meaning is sometimes expanded to somewhat earlier utterances. Last words of famous or infamous people are sometimes recorded (although not always accurately), which then became a historical an ...

to his younger brother Alfred were: On 2 June, Évariste Galois was buried in a common grave of the

Montparnasse Cemetery Montparnasse Cemetery () is a cemetery in the Montparnasse quarter of Paris, in the city's 14th arrondissement of Paris, 14th arrondissement. The cemetery is roughly 47 acres and is the second largest cemetery in Paris. The cemetery has over 35,00 ...

whose exact location is unknown. In the cemetery of his native town –

Bourg-la-Reine Bourg-la-Reine () is a Communes of France, commune in the southern suburbs of Paris, France. It is located from the Kilometre Zero, center of Paris. History In 1792, during the French Revolution, Bourg-la-Reine (meaning "Town of the Queen") w ...

– a

cenotaph A cenotaph is an empty grave, tomb or a monument erected in honor of a person or group of people whose remains are elsewhere or have been lost. It can also be the initial tomb for a person who has since been reinterred elsewhere. Although t ...

in his honour was erected beside the graves of his relatives. Évariste Galois died in 1832. Joseph Liouville began studying Galois's unpublished papers in 1842 and acknowledged their value in 1843. It is not clear what happened in the 10 years between 1832 and 1842 nor what eventually inspired Joseph Liouville to begin reading Galois's papers. Jesper Lützen explores this subject at some length in Chapter XIV ''Galois Theory'' of his book about

Joseph Liouville Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...

without reaching any definitive conclusions. It is certainly possible that mathematicians (including Liouville) did not want to publicize Galois's papers because Galois was a republican political activist who died 5 days before the June Rebellion, an unsuccessful anti-monarchist insurrection of Parisian republicans. In Galois's obituary, his friend Auguste Chevalier almost accused academicians at the École Polytechnique of having killed Galois since, if they had not rejected his work, he would have become a mathematician and would not have devoted himself to the republican political activism for which some believed he was killed. Given that France was still living in the shadow of the

Reign of Terror The Reign of Terror (French: ''La Terreur'', literally "The Terror") was a period of the French Revolution when, following the creation of the French First Republic, First Republic, a series of massacres and Capital punishment in France, nu ...

and the

Napoleonic era The Napoleonic era is a period in the history of France and history of Europe, Europe. It is generally classified as including the fourth and final stage of the French Revolution, the first being the National Assembly (French Revoluti ...

, Liouville might have waited until the political turmoil subsided (from the failed June Rebellion and its aftermath) before turning his attention to Galois's papers. Liouville finally published Galois's manuscripts in the October–November 1846 issue of the ''

Journal de Mathématiques Pures et Appliquées The ''Journal de Mathématiques Pures et Appliquées'' () is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874). The journal was originally published by Charles Louis Étienne Bachelier. ...

''. Galois's most famous contribution was a novel proof that there is no quintic formula – that is, that fifth and higher degree equations are not generally solvable by radicals. Although

had already proved the impossibility of a "quintic formula" by radicals in 1824 and

Paolo Ruffini Paolo Ruffini (22 September 1765 – 10 May 1822) was an Italian mathematician and philosopher. Education and career By 1788 he had earned university degrees in philosophy, medicine/surgery and mathematics. His works include developments in a ...

had published a solution in 1799 that turned out to be flawed, Galois's methods led to deeper research into what is now called

Galois Theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

, which can be used to determine, for ''any'' polynomial equation, whether it has a solution by radicals.

Contributions to mathematics

From the closing lines of a letter from Galois to his friend Auguste Chevalier, dated 29 May 1832, two days before Galois's death: 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. His work has been compared to that of

(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 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

. He called the decomposition of a group into its left and right

coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...

s a ''proper decomposition'' if the left and right cosets coincide, which leads to the notion of what today are known as

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

s. He also introduced the concept of a

(also known as a

Galois field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

in his honor) in essentially the same form as it is understood today. In his last letter to Chevalier and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields: *He constructed the 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^ν''. *He constructed the

projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...

PSL(2,''p''). Galois constructed them as fractional linear transforms, and observed that they were simple except if ''p'' was 2 or 3. These were the second family of finite

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

s, after the

alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...

s. *He noted the exceptional fact that PSL(2,''p'') is simple and acts on ''p'' points if and only if ''p'' is 5, 7, or 11.

Galois theory

Galois's most significant contribution to mathematics is his development of Galois theory. He realized that the algebraic solution to a

equation is related to the structure of a group of

permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...

s associated with the roots of the polynomial, the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

of the polynomial. He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, that is, its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the

to which Galois originally applied it.

Analysis

Galois also made some contributions to the theory of Abelian integrals and

continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...

s. As written in his last 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, Galois proved that the regular continued fraction which represents a quadratic surd ''ζ'' is purely periodic if and only if ''ζ'' is a reduced surd, that is,

\zeta > 1

and its conjugate

\eta

satisfies

-1 < \eta < 0

. 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 :

, \end

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'' > 1 is a rational number that is not a perfect square, then :

\sqrt = \left,a_0;\overline\,\right

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'' − 1 partial denominators form a

palindromic A palindrome ( /ˈpæl.ɪn.droʊm/) is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as ''madam'' or '' racecar'', the date " 02/02/2020" and the sentence: "A man, a plan, a canal – Pana ...

string.

Notes

References

* – Reprinting of second revised edition of 1944, The University of Notre Dame Press. * *. Still in print. * * – This textbook explains Galois Theory with historical development and includes an English translation of Galois's memoir. * * – Classic fictionalized biography by physicist Infeld. * * – This biography challenges the common myth concerning Galois's duel and death. * – This comprehensive text on Galois Theory includes a brief biography of Galois himself. * – Historical development of Galois theory. *

External links

* * *
The Galois Archive
(biography, letters and texts in various languages) * Two Galois articles, online and analyzed on ''BibNum'' : "Mémoire sur les conditions de résolubilité des équations par radicaux" (1830)
link
or English analysis, click 'A télécharger'/small>; "Démonstration d'un théorème sur les fractions continues périodiques" (1829)
link
or English analysis, click 'A télécharger'/small> *
La vie d'Évariste Galois by Paul Dupuy
The first and still one of the most extensive biographies, referred to by every other serious biographer of Galois * !-- http://perso.univ-rennes1.fr/antoine.chambert-loir/DJVU/ -->https://www.irphe.fr/~clanet/otherpaperfile/articles/Galois/N0029062_PDF_1_84.pdf Œuvres Mathématiquespublished in 1846 in the ''Journal de Liouville'', converted to Djvu format by Prof. Antoine Chambert-Loir at the University of Rennes.
Alexandre Dumas, Mes Mémoires
the relevant chapter of Alexandre Dumas' memoires where he mentions Galois and the banquet. *
Theatrical trailer of University College Utrecht's "Évariste – En Garde"
* {{DEFAULTSORT:Galois, Evariste 1811 births 1832 deaths People from Bourg-la-Reine École Normale Supérieure alumni Lycée Louis-le-Grand alumni 19th-century French mathematicians Group theorists French murder victims People murdered in Paris Duelling fatalities Deaths by firearm in France Deaths from peritonitis Burials at Montparnasse Cemetery French duellists