Evariste Galois (pronounced 'Gel-wa') has been hailed by many as the father of modern algebra. In his short lifetime, he did some phenomenal research work on mathematics and published many of his works. Galois worked out algebraic applications of finite groups, now known as Galois groups, and laid the foundations for the solvability of algebraic equations using rational operations and extraction of roots. It is beyond doubt that his mathematical work helped a great deal in the transformation of the theory of algebraic equations. Along with Norwegian mathematician Niels Abel, he proved the impossibility of solving general quintic equation and polynomial equations of higher degree, in terms of a finite number of rational operations and root extractions. Galois had to endure many misfortunes in his short lifetime, ranging from his father's untimely demise to many of his works being ignored, misplaced and lost by their caretakers.
Evariste Galois was born on the 25th of October, 1811 in Bourg-la-Rein, near Paris. Both of his parents were well educated in classical literature, religion and philosophy. Evariste’s father, Nicolas-Gabriel Galois, was a Republican and headed the Bourg-la-Reine's liberal party. Evariste’s mother, Adélaïde-Marie took care of Galois’s education till he turned twelve when he entered the lycée of Louis-le-Grand in Paris in October 1823. Though the school was going through a great upheaval when Galois entered and about 100 students were expelled, he performed well initially and ranked first in Latin which he learnt under his mother’s tutelage. However, he soon lost interest in studies and started taking deep interest only in mathematics, at the age of 14. He found a copy of Adrien Marie Legendre's Éléments de Géométrie, which it is said that he read "like a novel" and mastered at the first reading. It must be mentioned that even now professional mathematicians find this book too difficult to master. He had no patience or need of studying "pupils"; he was in search of "masters" ! At 15, he was reading the original papers of Joseph Louis Lagrange, such as the landmark Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory.
At age 16, Galois took the examination to enter the prestigious École Polytechnique, but failed. Bell and others blame the failure, at least in part, on the arrogance of a young man who did not suffer fools well, and fools were what he considered his examiners to be. They put questions to him about mathematics he considered trivial. He wished to demonstrate his abilities by describing the problems he had studied and the mathematics he had invented to solve them. His examiners were not interested, insisting instead that he answer their questions. His halfhearted attempts to do so were not impressive. It has been reported that Galois was never very good at communicating his thoughts, perhaps because to him things were too clear to need elaboration. 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.
Galois published his first mathematics paper on continued fractions in the Annales de mathématiques in April, 1829. Around this time, he was also studying the theory of polynomial equations on which he submitted two papers, on 25 May and 1 June, to the Académie des Sciences. Augustin Louis Cauchy, an eminent mathematician of the time, evaluated Galois's paper but refused to publish it for reasons which still remain obscure.
On 28 July 1829 Galois's father committed suicide after a bitter political dispute with the village priest.
Galois made another attempt to enter the Polytechnique but failed yet again. However, different accounts suggest different reasons behind his failure. Some suggest that, the exercise which he was given by the examiner was of very little interest, which annoyed him to the core and he threw the blackboard cleaning rag at the examiner. According to more popular accounts, Galois made too many logical leaps which were hard to grasp for the examiner, and this infuriated Galois. But, it is believed that the tragic death of his father was the chief reason behind his odd behavior. A candidate of superior intelligence is lost with an examiner of inferior intelligence ! Eric Temple Bell, the famous historian of mathematics in his book “Men of Mathematics” quotes
"Men who were not worthy to sharpen his pencils sat in judgement on him "
After failing to secure a seat at the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He cleared with a degree on the 29th of December 1829. His mathematics examiner reported, “This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research.” On the other hand his literature examiner reported, “This is the only student who has answered me poorly, he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, for, after his examination, I believed him to have but little intelligence.”
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. 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.
During Galois’s time, France was going through a great political unrest. In July 1830, when the director of the École Normale, M. Guigniault, locked-in his students to prevent them from participating in rioting, Galois wrote a letter criticizing Guigniault and was consequently expelled from École Normale on the 4th of January 1831. Galois was affiliated to republican organisations like Republican Artillery of the National Guard and the Society of the Friends of the People and split his time between mathematics and politics.
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. 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's work "incomprehensible", declaring that "(Galois's) 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's 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.
In less than a month after being released from prison, Galois was challenged to a duel. Galois's fatal duel took place on 30th May, 1832. Though the reason behind the duel is not clear, there have been a great many speculations. Some letters written prior to his death can be traced back to a woman named Mademoiselle Stéphanie-Félicie Poterin du Motel, who might have shared some of her personal problems with Galois and this could have instigated the duel.
The events of the night before the duel have been the source of much speculation. Here’s how Bell’s delicious prose set the scene:
“All night long he had spent the fleeting hours feverishly dashing off his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death which he saw could overtake him. Time after time he broke off to scribblein the margin ‘I have not time; I have not time,’ and passed on to the next frantically scrawled outline. What he wrote in those desperate last hours before dawn will keep generations of mathematicians busy for hundreds of years.”
Image : Notes from Galois last pages before his death
As the legend, goes the next morning he appeared at the dueling grounds, was fatally wounded and abandoned by his opponent and even his own seconds. He was discovered by a peasant and taken to a hospital. When his younger brother Alfred arrived, Évariste told him, in fact this were his last words,
“Don’t cry Alfred, I need all my courage to die at twenty.”
He died the day after the duel at the age of 20 !!! . On 2 June, Évariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown. In the cemetery of his native town – Bourg-la-Reine – a cenotaph in his honour was erected beside the graves of his relatives.
It would make marvelous drama, but bad mathematics to believe that it was the night before the duel, in the state he must have been, that he created his marvelous theory of equations. Galois had already written memoirs on his use of groups in his work in the theory of equations. In a long letter to Auguste Chevalier, written that night, Galois described his theory and the contents of the memoir rejected by Poisson.
Alfred and Chevalier copied his mathematical papers and sent them to Gauss, Jacobi and others as Évariste had requested in his last letters. No record exists of any comment these men made.
Galois's 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 Mathématiques Pures et Appliquées. And then his legend began.
As so much about Galois’ life and death is in question it is fitting to end this account with something certain, his mathematical legacy. The solution of quadratic equations goes back to ancient people, including the Babylonians, Chinese, and Hindus. Girolamo Cardano published formulas in 1545 for solutions of the cubic and quartic equations, after Niccolo Tartaglia and Ludovico Ferrari discovered them a few years earlier. Mathematicians were unable to find formulas involving radicals for solving quintic degree equations. In 1796 Ruffini attempted to prove this was impossible, but his efforts were not wholly successful. In 1824 Niels Abel gave an essentially correct proof. Galois was unaware of Abel’s work when he began his own investigation. Galois's methods led to deeper research in what is now called Galois theory. He proved that no general method using purely algebraic formulas could be found to solve equations of degree five or higher His results became a major factor in the evolution from classical to modern algebra, from the solving of equations to the study of systems.
"You will publicly ask Jacobi or gauss to give their opinion not on the truth, but on the importance of the theorems. after this there will be, I hope, some people will find it to their advantage to decipher all this mess" – Évariste Galois, Letter to Auguste Chevalier, May 29, 1832.
Written by Dhiraj Sarmah on 31st May, 2013.