Scientists



GALOIS
Evariste Galois (1811 – 1832) was a French mathematician born in BourglaReine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals,
thereby solving a longstanding problem. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra, and the subfield of Galois connections. He was the first to use the word
"group" as a technical term in mathematics to represent a group of permutations. A radical Republican during the monarchy of Louis Philippe in France, he died from wounds suffered in a duel under questionable circumstances
at the age of twenty.
Galois was born on 25 October 1811 to NicolasGabriel Galois and AdelaideMarie (born Demante). His father was a Republican and was head of BourglaReine's liberal party. He became mayor of the village after Louis XVIII
returned to the throne in 1814. His mother, the daughter of a jurist, was a fluent reader of Latin and classical literature and was responsible for her son's education for his first twelve years. At the age of 10, Galois
was offered a place at the college of Reims, but his mother preferred to keep him at home.
In October 1823, he entered the Lycee LouisleGrand, and despite some turmoil in the school at the beginning of the term (when about a hundred students were expelled), Galois managed to perform well for the first two
years, obtaining the first prize in Latin. He soon became bored with his studies and, at the age of 14, he began to take a serious interest in mathematics.
He found a copy of Adrien Marie Legendre's The Elements of Geometry, which it is said that he read "like a novel" and mastered at the first reading. At 15, he was reading the original papers of Joseph Louis Lagrange,
such as the landmark Reflexions sur la resolution algebrique des equations which likely motivated his later work on equation theory, and Lecons sur le calcul des fonctions, work intended for professional
mathematicians, yet his class work remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.
In 1828, he attempted the entrance examination for the Polytechnic School, 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 Normal School, a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.
In the following year Galois's first paper, on continued fractions, was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers
on this topic to the Academy of Sciences. Augustin Louis Cauchy 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' 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, considered Galois' work to be a likely winner.
On 28 July 1829 Galois's father committed 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 Polytechnic School, and failed
yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. The legend holds that he thought the exercise proposed to him by the examiner to be of no interest, and, in
exasperation, threw the rag used to clean the blackboard at the examiner's head. 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 Polytechnic School, Galois took the Baccalaureate examinations in order to enter the Normal School. 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".
He submitted his memoir on equation theory several times, but it was never published in his lifetime due to various events. As noted before, his first attempt was refused by Cauchy, but in February 1830 following Cauchy's
suggestion he submitted it to the Academy's secretary Fourier, 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
Abel posthumously and also to Jacobi. Despite the lost memoir, Galois published three papers that year, one of which laid the foundations for Galois theory. The second one was about the numerical resolution of equations
(root finding in modern terminology). The third was an important one in number theory, in which the concept of a finite field was first articulated.
Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII 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 abdication, staged a coup d'etat, and issued his notorious July Ordinances, touching off the July Revolution which ended with LouisPhilippe becoming king. While their counterparts at the Polytechnic
School were making history in the streets during les Trois Glorieuses, Galois and all the other students at the Normal School were 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 Ecoles, 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. 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' former unit were arrested and charged with conspiracy to overthrow the government.
In April, 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. The proceedings grew riotous, and Galois proposed
a toast to King LouisPhilippe with a dagger above his cup, which was interpreted as a threat against the king's life. He was arrested the following day, but was acquitted on 15 June.
On the following Bastille Day, Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. For this, he was again arrested and
this time was sentenced to six months in prison for illegally wearing a uniform. He was released on 29 April 1832. During his imprisonment, he continued developing his mathematical ideas.
Galois returned to mathematics after his expulsion from the Normal School, 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. Simeon Poisson asked him to submit his work on the theory of equations, which he did
on 17 January. Around 4 July, Poisson declared Galois' work "incomprehensible", declaring that "[Galois'] argument 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' Bastille Day
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, began collecting all his mathematical manuscripts while
still in prison, and continued polishing his ideas until his release on 29 April 1832.
Galois' fatal duel took place on 30 May. The true motives behind the duel will most likely remain forever obscure. There has been much speculation, much of it spurious, as to the reasons behind it. 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 a Mademoiselle StephanieFelicie 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 either obliterated, such as her name, or deliberately omitted) are available. The letters hint that Mlle 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. Much more detailed speculation based on these scant historical details has been interpolated by many of Galois' biographers (most notably by Eric Temple Bell in Men of Mathematics), such as the frequently repeated
speculation that the entire incident was stagemanaged by the police and royalist factions to eliminate a political enemy.
As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville, one of the nineteen artillery officers whose acquittal was celebrated at the banquet that occasioned Galois' first arrest and du Motel's fiance.
However, Dumas is alone in this assertion, and extant newspaper clippings from only a few days after the duel give a description of his opponent that more accurately applies to one of Galois' 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. Hermann Weyl, a mathematician, 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.
Early in the morning of 30 May 1832 he was shot in the abdomen and died the following morning at ten o'clock in the Cochin hospital (probably of peritonitis) after refusing the offices of a priest. He was 20 years old.
His last words to his brother Alfred were: "Don't cry, Alfred! I need all my courage to die at twenty".
On 2 June, Evariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown. In the cemetery of his native town – BourglaReine – a cenotaph in his honor was erected beside the
graves of his relatives.
"Ask Jacobi or 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"
– wrote Galois to his brother Alfred and his friend Auguste Chevalier. They have sent last works of Galois to Gauss and Jacobi, but have not waited the answer.
Galois' mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the "Journal de Mathematiques
Pures et Appliquees". The most famous contribution of this manuscript was a novel proof that there is no quintic formula. That is, fifth and higher degree equations those are not solvable by radicals. Although Abel had
already proved the impossibility of a "quintic formula" by radicals in 1824 and Ruffini had published a solution in 1799 that turned out to be flawed, Galois' methods led to deeper research in what is now called Galois
theory. For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.
Unsurprisingly, Galois' collected works amount to only some 60 pages, but within them are many important ideas that have had farreaching consequences for nearly all branches of mathematics. His work has been compared to
that of Niels Henrik Abel, another mathematician who died at a very young age, and much of their work had significant overlap.
While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group 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 developed the concept that is today known as a normal subgroup. He called the decomposition of a group into its left and right cosets a
proper decomposition if the left and right cosets coincide, which is what today is known as a normal subgroup. 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.
Galois' most significant contribution to mathematics by far 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 permutations
associated with the roots of the polynomial, the Galois group 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, or its Galois group is 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.
Galois also made some contributions to the theory of Abelian integrals and continued fractions.
