Évariste Galois


French mathematician (1811-1832) who made important contributions to mathematics, including group theory.

For further interest, a biography follows below.

Brilliance in a Short and Tragic Life

variste Galois was born in the village of Bourg-la-Reine near Paris on October 25, 1811 to Nicholas Gabriel Galois and Adelaide Marie Galois Demante. Both of his parents were intelligent and well educated in philosophy, classical literature and religion. However, there is no indication of any special mathematical ability in any members of Galois' family. His mother was his sole teacher until he was 12 years old, teaching him subjects such as Greek, Latin and religion, where she imparted her own skepticism to her son. Galois' father was an influential member of the community and in 1815, he was elected mayor of the village.

Galois enrolled at the Lycée of Louis-le-Grand in Paris as a boarding student in October, 1823. During 1824-25, his school record was good and he received several prizes. However in 1826, Galois was asked to repeat the year because his work in rhetoric was not up to the required standard.

February, 1827, was a turning point in Galois' life. He enrolled in his first mathematics class, the class of M. Vernier. He quickly became absorbed in mathematics and his director of studies wrote:

It is the passion for mathematics which dominates him, I think it would be best for him if his parents would allow him to study nothing but this, he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments.

Galois' school reports began to describe him negatively as singular, bizarre, original, and closed . It is interesting that perhaps one of the most creative mathematicians who ever lived should be criticized for being original. However, M. Vernier did evaluate him as having:

Intelligence, marked progress, but not enough method.

In 1828, Galois took the examination to enter the École Polytechnique, the leading technical institute in Paris, but failed. Many years later, Terquem noted:

A candidate of superior intelligence is lost with an examiner of inferior intelligence.

Galois had wished to enter this school for academic reasons, but he also desired to enter it because of the strong political movements that existed among its students, since he followed his parents' example in being an ardent republican. Back at Louis-le-Grand, Galois enrolled in the mathematics class of Louis Richard. However he worked more on his own research than on his schoolwork. He studied Legendre's Géométrie and the treatises of Lagrange. As Richard was to report:

This student works only in the highest realms of mathematics.

In April, 1829, Galois had his first mathematics paper published on continued fractions in the Annales de Mathématiques . In May and June of the same year, he submitted articles on the algebraic solution of equations to the Acadiémie des Sciences. Cauchy was appointed as referee of Galois' paper.

Tragedy struck Galois in July, 1829, as his father committed suicide. The politically motivated priest of Bourg-la-Reine forged the mayor Galois' name on malicious epigrams directed at Galois' own relatives. Galois' father was a good-natured man and the scandal that ensued was more than he could stand. He hanged himself in his Paris apartment only a few steps from Louis-le-Grand where his son was studying. Galois was deeply affected by his father's death and it greatly influenced the direction his life was to take.

A few weeks after his father's death, Galois presented himself for examination for entry to the École Polytechnique for the second time. Again he failed, no doubt partly due to the recent death of his father, and partly because he was never adept at communicating his deep mathematical ideas. Galois therefore resigned himself to enter the École Normale, which was an annex to Louis-le-Grand, and to do so he had to take his Baccalaureate examinations, a task he could have avoided by entering the École Polytechnique.

He passed, receiving his degree on December 29, 1829. His examiner in mathematics stated:

This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research.

In startling contrast, 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.

Undeterred, Galois continued his research and sent Cauchy further work on the theory of equations, but then learned from the Bulletin de Férussac of a posthumous article by Abel which overlapped with a part of his work. On Cauchy's advice, Galois submitted a brilliant new article on the condition that an equation be soluble by radicals in February 1830. The paper was sent to Fourier, the secretary of the Académie des Sciences, to be considered for the Grand Prize in mathematics. Sadly, Fourier died in April 1830 and Galois' paper was misplaced during the resulting reorganization at the Académie.

The Grand Prize was awarded in June, jointly to Abel (posthumously) and to Jacobi, for their work on elliptic functions and abelian integrals. Galois' paper had never been considered for the prize due to the unfortunate timing of Fourier's death. Coincidently, Galois had published a few papers several weeks earlier before the prize, theoretically extending Abel and Jacobi's work himself. It appears this feat was also not duly recognized until after many years later.

July of 1830 saw massive public uprisings and the king, Charles the 10th, fled France. There was rioting in the streets of Paris and the director of École Normale, M. Guigniault, locked the students in to prevent them from taking part. Galois, with his strong republican beliefs, tried to scale the wall to join the riots but failed. In early December, 1830, M. Guigniault wrote newspaper articles attacking the students and Galois wrote a reply in the Gazette des Écoles, attacking M. Guigniault for his actions in locking the students into the school. For this letter Galois was expelled, whereupon he joined the Artillery of the National Guard, a republican branch of the militia. However, shortly thereafter, in late December, 1830, this militia branch was abolished by royal decree since the new king, Louis-Phillipe, felt it was a threat to the throne.

Two minor articles, an abstract in Annales de Gergonne (December, 1830) and a letter on the teaching of science in the Gazette des Écoles ( January 1831) were Galois' last publications during his lifetime. In early January, 1831, Galois attempted to return to mathematics. He organized some mathematics classes in higher algebra which attracted 40 students to the first meeting but after that the numbers quickly fell off. Galois was invited by Poisson to submit another version of his paper on equations to the Académie and he did so in mid-January.

In April, acquaintance Sophie Germain wrote a letter to her friend the mathematician Libri which tells of Galois' condition:

... the death of M. Fourier, have been too much for this student Galois who, in spite of his impertinence, showed signs of a clever disposition. All this has done so much that he has been expelled from École Normale. He is without money... They say he will go completely mad. I fear this is true.

In May, 1831, Galois was arrested, after a republican dinner gathering, because he had purportedly made vague threats towards the monarchy. Despite being acquitted in a trial, he was later rearrested on Bastille Day, July 14, for wearing the illegal uniform of the defunct Artillery of the National Guard. He was sent to Sainte-Pélagie prison, which was where he learned of Poisson's rejection of his latest work. Poisson believed:

His argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigour.

However, Poisson did encourage Galois to publish a more complete account of his work. Deeply troubled with his situation, Galois attempted to commit suicide by stabbing himself with a dagger but the other prisoners prevented him.

In March, 1832, a cholera epidemic swept Paris and prisoners, including Galois, were transferred to the pension Sieur Faultrier. There he apparently fell in love with Stephanie-Felice du Motel, the daughter of the resident physician. After he was released in late April, Galois exchanged letters with Stephanie, and it is clear that she tried to distance herself from him. The name Stephanie appears several times as a marginal note in one of Galois' manuscripts.

In late May, Galois was challenged to a pistol duel at 25 paces by Perscheux d'Herbinville, a rival in politics and also for Stephanie's affection. Legend has it that Galois stayed up the entire night before the fateful duel, feverishly writing what would become his final notes on mathematics, especially pertaining to group theory.

On May 30, 1832, Galois was shot in the abdomen in the duel and left to die in a field by the victorious d'Herbinville. Found by a peasant, Galois succumbed to his wounds the next day in a hospital.

He was 20 years old.

In 1846, Liouville editted and published Galois' manuscripts, including his final notes. Liouville announced to the Académie des Sciences:

I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem: In order that an irreducible equation of prime degree be solvable by radicals it is necessary and sufficient that all its roots be rational functions of any two of them.

Galois finally began receiving his redemption and recognition, many years after the end of his very short life.

The immensely important contributions to the field of mathematics by Évariste Galois are now fully and justly recognized.

His work holds tremendous influence to this day.