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Evariste Galois

Clint Shaffer

25 March 2007

History of Math

Math 4010

Dr. Bill Cherowitzo Shaffer 2

Evariste Galois was a brilliant mathematician, famous for his work with algebraic equations and quadratic theory. Galois’ father Nicholas-Gabriel Galois was the mayor of the town where Evariste was born, Evariste’s mother Adelaide- Marie Demante was from a family of legal professionals and was educated in classic cultures by her father.

Evariste’s mother was his only teacher for the first twelve years of his life and when

Evariste finally did attend a public institution of education, it was comparable to a prison with a strict regimen facilitated by the faculty at Lycee Louis-le-Grand.

The school was famous for its classical studies, but the militant daily schedule and horrible living conditions made it difficult for Evariste to tolerate. While repeating his third year at Lycee Louis-le-Grand, Evariste enrolled into the class Preparatory

Mathematics; the teacher, Mr. Hippolyte Vernier introduced a new textbook, Elements of Geometry by Legendre that changed Evariste’s life. Evariste began to grow bored of the curriculum at Lycee Louis-le-Grand and decided to take the entrance examination for Ecole polytechnique a year early; however, he failed and as a result of his first failure he made a second attempt at the entrance examination a year later but also failed this attempt rendering himself inadmissible. As a consequence of not being accepted into

Ecole polytechnique, Evariste decided to apply for admission into the less prestigious

Ecole preparatoire.

Evariste published his first mathematical research paper in 1829 at the age of seventeen. Politics had always played a significant role in Evariste’s life; his father’s teachings of republicanism were instilled in him when he was a young child. Evariste Shaffer 3 could not stand by passively and watch as the civil liberties of one his instructors at the school were attacked by the headmaster. It has been speculated that Evariste wrote a letter to the school newspaper, La Gazette des Ecoles, invoking controversy that was directed towards the headmaster and his relationship with the government. The editors of the paper deliberately removed the signature from the letter they received when it was published and Galois never confirmed nor denied that he had composed the letter; as a consequence of Galois’ supposed involvement in the letter, he was expelled immediately. Evariste enlisted into the National Guard, artillery division, but not much time had passed before his unit was disbanded. Evariste then decided to publicly voice his political views and led a six hundred person protest which resulted in him and his political partner of the time being arrested and serving a six month prison term.

Throughout Galois’ life he had many intense emotional stages, yet at such a young age he accomplished so much in the field of advanced mathematics. Galois was a mathematical genius wise beyond his years; he had a prophetic like sense in respect to his death and the night before he died Evariste wrote down what is now known as

Galois Theory, unfortunately Evariste Galois died at the youthful age of twenty, most likely far before his natural time.

Evariste Galois was born on the 25th of October in 1811 in a town called Bourg- la-Reine located near Paris, France. Evariste was named for the Catholic saint whose feast day fell on October 26th, the day after his birthday. Galois’ father, Nicolas-Gabriel

Galois was a liberal thinker and was appointed mayor of Bourg-la-Reine; Nicholas received the position of mayor under the second Restoration (Gillispie p. 260).

Evariste’s mother Adelaide- Marie Demante was an educated woman, she came from a Shaffer 4 family of jurists (legal experts), and was mostly taught by her father who was a jurisconsult in the Paris Faculty of Law (Livio p. 112). Adelaide taught her son Evariste for the first twelve years of his life, she stressed the importance of a traditional education for her children offering them a strong background in the classics and in the theology while at the same time instilling in them liberal ideology (Livio p. 114). When

Evariste was ten years old his mother sent him to school in Reims on a partial grant, but soon changed her mind and decided that he was too young to go to school so far away and instead decided to tutor him at home for two additional years (Infeld p. 4).

Evariste entered public school for a second time in 1823, when he attended Paris’ prestigious Lycee Louis-le-Grand at age 12. The school had a celebrated reputation for its instruction in classical studies; even though the atmosphere at the school was more reminiscent of a prison than of an institution of learning. Lycee Louis-le-Grand had become so prominent at the end of the seventeenth century that King Louis XIV decided that the establishment was worthy enough to bare his name (Rigatelli p. 21).

The typical schedule at the school was painstakingly strict and militant. At the school Evariste’s day started at 5:30am where he would don his uniform, designed by

Napoleon himself, complete with a two-cornered hat. The classrooms he attended were dark and dingy, rats could be seen running around in the classrooms and on the stairs.

The students were forced to sit on steps which served as desks in the institution; everything in the school was lit only by candlelight. In order to invoke a strong sense of authority the teacher’s desks were elevated high above the students, everything was done in silence; pupils had to dress, bath, and eat every meal, which usually consisted of water and dry bread, in silence (Rigatelli p. 22). There were around 500 pupils Shaffer 5 attending the school and even though the militant daily schedules and facade of the school gave the appearance that everything was under control, within the walls of the school the students were disobedient. Much like prison each infraction a student committed was punishable by forced solitary confinement in one of twelve punishment cells located on the campus of the school. However, rigorous the schedule, Evariste was still able to do well his first year at the school (Rigatelli p. 23-24). During Evariste’s first year at the school France’s political climate was darkening and controversy was surrounding the school, there was an unbalanced power triangle between, the church, the royalists, and the republicans.

As well as Evariste did his first year, time was progressing and with every year that past he grew more and more jaded of the curriculum. In Evariste’s third year at the school there was a restructuring and the ultraconservative Pierre-Laurent Laborie came into power at the school as the new headmaster (Livio p.116). With Evariste’s dampening respect for his current school system, Laborie did not favor Evariste and decided that Evariste was too young to advance on to his forth year of school; as a blessing in disguise, under Laborie’s discretion, Evariste was forced to repeat his third- year classes (Livio p. 116). This ended up not being such a bad situation for Evariste this was the year that Galois the mathematician was born. That same year there was a new Preparatory mathematics teacher at the school, Mr. Hippolyte Vernier. Vernier introduced a new math book to his classes entitled, Elements of Geometry by Legendre; it has been said that Evariste digested the book, intended for a full two-year course, in only two days and while this is highly suspect it is one of the stories that adds to the appeal of Galois’ legend (Livio p. 117). Shaffer 6

By the fall of 1827 Galois immersed himself into mathematics and ignored all of his other subjects, he refused to use any of the standard textbooks from the school and jumped right into studying original research papers Evariste read through every mathematical article of the time that he could get his hands on and fell head first into

Legendre’s memoirs, Resolution of Algebraic and Theory of Analytic Functions (Livio p.

117). Through reading Legendre’s books, Evariste decided to take on a grand task that other great mathematicians had also been working on; he wanted to solve the general quintic.

Evariste was unaware of two other mathematicians, Abel and Ruffini who had previously worked on the same problem. Galois worked on the problem for two months and just like Abel, Evariste thought that he had found the formula realizing later that there was an error in his solution. While Evariste was investigating the problem further he realized that the error, which he himself had made with the solution to the quintic, mirrored Abel’s mistake. Evariste, like Abel, was compelled to study the solvability of algebraic equations having realized his mistake and showed that there is no general solution for the quintic (Livio p. 117).

Evariste’s math teacher Mr. Vernier described him as being a genius in mathematics; however, Evariste neglected his other school subjects which in turn made his work lack methodology and unsystematic (Livio p. 118). Vernier tried to help

Evariste organize his works; however, Evariste ignored Vernier’s advice and decided to take the entrance exam for the prestigious Ecole polytechnique University a year early.

Evariste took the entrance exam in June of 1828 and failed, most likely due to his lack Shaffer 7 of preparation, Galois dreams of attending the prestigious school were fading (Livio p.

118). Evariste returned to Lycee Louis-le-Grand for another year where he enrolled into

Louis-Paul-Emile Richard’s “Special Mathematics” class, Richard was a exceptionally enthusiastic and supportive teacher for Galois and became one of Galois’ first mentors

(Livio p. 118).

In 1829 Galois published his first research paper, it entailed research done with continued fractions, and quadratic theory, it was published in the journal Annales de

Mathematiques pures et appliqués (Livio p. 118). At seventeen years old Evariste

Galois was about to revolutionize algebra. In Abel’s attempt of solving the quintic he came to the conclusion that the problem could not be solved with a formula, Galois having also come to this conclusion branched out on his own research creating his own theory regarding the seminal concept of a group and establishing a new form of algebra which is now known as Galois Theory. It is in this approach that the brilliance of Galois can be seen, this method “provided a group-theoretic criterion for the solution of an equation by radicals” which in turn “led to the modern-day Galois Theory” (Gallian p.

555). It has been stated that Evariste Galois is the father of modern algebra; before

Galois algebraists were for the most part concentrating their efforts on the general solution of polynomial equations. Scipione dal Ferro, Tartaglia, and Cardano expressed how to solve cubic equations, and Ferrari expressed how to solve the “biquadratic”

(Waerden p. 76). It should be noted that Evariste expressed the notion of solvability by radicals in terms of the property of the group and not by the properties of the equation

(Grattan p. 718). By applying his method, Evariste was able to associate with each Shaffer 8 equation the “Galois” group of the equation, where the Automorphism, the Galois group and the Fixed Field of H are defined by the following.

Definition- Automorphism, Galois group, Fixed Field of H- “Let E be an

extension field of the field F, an automorphism of E is a ring isomorphism from E

onto E. The Galois group of E over F, Gal(E/F), is the set of all automorphisms of

E that take every element of F to itself. If H is a subgroup of Gal(E/F), the set

EH = {x є E : Φ(x) = x for all Φ є H}

Is called the fixed field of H” (Gallian p. 548).

Now with these terms defined the Fundamental Theorem of Galois Theory can be stated.

Fundamental Theorem of Galois Theory- “Let F be a field of characteristic 0 or

a finite field. If E is the splitting field over F for some polynomial in F[x], then the

mapping from the set of subfields of E containing F to the set of subgroups of

Gal(E/F) given by K → Gal(E/K) is a one-to-one correspondence. Furthermore,

for any subfield K of E containing F

1. [E: K] = │Gal(E/K)│ and [K: F] = │Gal(E/F)│/│Gal(E/K)│. (The index of

Gal(E/K) in Gal(E/F) equals the degree of K over F.)

2. If K is the splitting field of some polynomial in F[x], then Gal(E/K) is a

normal subgroup of Gal(E/F) and Gal(K/F) is isomorphic to

Gal(E/F)/Gal(E/K). Shaffer 9

3. K = EGal(E/K). (The fixed field of Gal(E/K) is K.)

4. If H is a subgroup of Gal(E/F), then H = Gal(E/EH). (The automorphism

group of E fixing EH is H.)” (Gallian p. 552).

It was this method of approach that allowed Evariste to determine whether or not an equation is solvable by a formula or not. By analyzing the properties of the “Galois” group, Evariste was able to determine if a polynomial of nth degree was solvable by radicals. Now, “a polynomial in F[x] is solvable by radicals… if each root of the polynomial can be written as an expression involving elements of F combined by the operations of addition, subtraction, multiplication, division, and extraction of roots”

(Gallian p. 555). Or equivalently defined mathematically as,

Definition- Solvable by Radicals- “Let F be a field, and let f(x) є F[x]. We say

that f(x) is solvable by radicals over F if f(x) splits in some extension

F(a1, a2, …, an) of F and there exists positive integers k1, …, kn such that

a1^(k1) є F and ai^(ki) є F(a1, a2, …, ai-1) for i = 2, …, n” (Gallian p. 555).

Furthermore, a solvable group is defined as,

Definition- Solvable Group- “We say that a group G is solvable if G has a

series of subgroups {e} = H0 H1 H2 Hk = G, where, for each

0 ≤ i < k, Hi is normal in Hi+1 and Hi+1/Hi is Abelian.” (Gallian p. 556).

Therefore, the problem of solving a polynomial of nth degree can be changed into a problem about field extensions and by applying the Fundamental Theorem of Galois

Theory, the problem of field extensions can be transformed into a problem about groups Shaffer 10

(Gallian p. 555). It was by applying this method that Galois was able to show that there exists fifth degree polynomials that cannot be solved by radicals. In modern terms this method states that an “equation is solvable by radicals if and only if its Galois group is a

‘solvable group’” (Grattan p. 718).

Richard kept twelve of Galois’ notebooks containing his mathematical classwork; and in 1829 Richard encouraged Evariste to publish some in the form of two memoirs.

Richard himself was prepared to submit Evariste’s manuscripts to Cauchy, so that

Cauchy could present them to the Academy of Science. Undeniably the memoirs were submitted on May 25 and June 1 of 1829, entrusted to Cauchy, Joseph Fourier, Claude

Navier and Denis Poisson for analysis and judgment (Livio p. 119). Over six months later on January 18, 1830 Cauchy wrote to the academy stating that, due to being

“indisposed at home” he would be unable to present the work of Galois; and during the next session on January 25, 1830 Cauchy never mentioned Galois work (Livio p. 120).

In February of 1830, with a few modifications Evariste submitted his memoirs to the

Academy of Science as an entry for a prize. The prize committee consisted of the well- known mathematicians Poinsot, Lacroix, Poisson, and Legendre; however, for some unknown reason Fourier, the academy’s secretary took Galois’ work home. Fourier died on May 16 and Galois’ work was never recovered for judgment in the contest (Livio p.120). Evariste, having learned of the mistake that had resulted in his paper not even being considered for the prize became convinced that there was a conspiracy to deny him his just recognition. Shaffer 11

Besides Evariste’s troubles with his manuscripts the year 1829 was plagued with sadness for young Evariste. A new young priest came into power in Bour-le-Reine and sided with right wing administrators who succeeded in forcing Evariste’s father, Nicolas-

Gabriel Galois out of power. The priest forged Nicholas-Gabriel’s signature on some documents; as a result his father was forced to step down from his mayor ship back in

Evariste’s hometown of Bourg-la-Reine and due to his humiliation over the scandal of loosing his job Nicolas committed suicide on the second of June by gas asphyxiation

(Livio p. 120-121). The death of his father was a tremendous blow to Evariste and regrettably at his father’s funeral a riot broke out when the priest that was responsible for Nicholas-Gabriel’s demise attempted to participate in the funeral services (Livio p.

121). Evariste was experiencing of the worst time of his life, yet still determined to further his education he went to Ecole polytechnique for a second time to take the entrance exam.

The two examiners Charles Louis Dinet and Lefebure de Fourcy were described by the historian E.T. Bell, as “not worthy enough to sharpen his (Galois) pencil” (Livio p.

121). As described by Mario Livio in his book, The Equation That Couldn’t Be Solved:

How Mathematical Genius Discovered the Language of Symmetry, both Diney and

Fourcy are best known today for failing one of the greatest mathematical geniuses of all time (Livio p. 121). History has not recorded exactly what happened to Galois during his second examination at Ecole polytechnique but it has been speculated that Evariste became so agitated during the oral portion of the exam that he picked up a blackboard eraser and threw it at the examiners (Livio p. 122). Failing for a second time meant that Shaffer 12

Evariste was inadmissible to the school and he was forced to find another route of education.

Evariste having no other options available to him was required to attend the less prestigious Ecole preparatoire if he desired to continue his education; once again

Evariste had to take a series of examinations in order to attend. Evariste excelled at the mathematics portion and essentially that is what secured him a space as a pupil in the school; however, one of the physics examiners Jean Claude Peclet had scrutinized

Evariste writing that “he knows absolutely nothing…I have been told that he great at mathematics. This greatly surprises me” (Livio p. 122). He was admitted into Ecole preparatoire at the beginning of 1830 with a major in sciences (Divio p. 122). That same year Galois published three research papers, two on equations and one on the theory of numbers (Livio p. 123).

During this time of Evariste’s life, French politics were ever-changing, King

Charles X was in control and a strong opposition was forming, there were two parties who were against his rule. The opponents were divided into two different parties the republicans and the Orleanists (Livio p. 124). As Evariste was attending school at Ecole preparatoire, rebellions were taking place right outside of the school’s walls. M.

Guigniault the director of Ecole preparatoire threatened to call in armed troops to prevent the students from joining in rebellion. Evariste grew frustrated being stuck behind the walls of his school; his republican ideals rooted by the teachings of his father and mother compelled him to want to take part in the revolution, he tried to scale the school’s walls several times but too no avail. After three days and almost four thousand Shaffer 13 people had lost their lives a compromise was finally reached between the two opposing sides quelling the rebellion (Livio p. 124-125).

When Evariste returned home to Bourg-la-Reine after his first year at Ecole preparatoire he was a changed man; his surviving family was surprised to find that

Evariste was now an idealistic revolutionary. Upon returning to school in the fall he joined up with a militant wing of the republican party called, Societe des Amis du Peuple

(Society of the Friends of the People); according to Livio “the society had a reputation for not hesitating to use aggressive and even violent means to achieve its goals” (Livio p. 125). M. Guigniault sent a letter to one of the two school newspapers attacking one of the liberal instructors at Ecole preparatorie this along with his statement declaring that “good students should not be interested in politics” potentially irked Evariste (Livio p. 126). The rebuttal came promptly with a letter sent to, La gazette des ecoles (The

Schools’ gazette) criticizing Guigniault for his actions during the rebellion as well as for his, “narrow outlook and ingrained conservatism” (Livio p. 127). The newspaper editors deliberately removed the signed name at the bottom of the letter before publishing it, and as it would happen, Evariste never admitted nor denied composing the letter. After the letter went public, with no proof or evidence Guigniault expelled Evariste without delay claiming that he was tired of putting up with Evariste’s disobedience and his expulsion from the school was justified (Rigatelli p. 72). Galois had no support from the student body and the art students from the same institution even publicly sided with the decision. The school’s newspapers kept printing letters surrounding the incidence and

Evariste’s last appeal was printed on December 30th in it he stated, “I am not asking Shaffer 14 anything for myself, but speak out for your own honor and according to your conscience” (Livio 128).

After having been expelled from Ecole preparatorie Evariste enlisted in the artillery of the national guard; Evariste’s unit was disbanded but he kept his uniform and continued to wear it after the reorganization of the national guard (Livio p. 129).

Evariste’s life was beginning to unravel; his political reputation doomed him from any perspective jobs and banished him from most events (Livio p. 130). On April 16, 1831 a celebration banquet was being held in honor of nineteen disbanded artillery of the national guard members that were acquitted from a major trial; the banquet was being held at a well-known restaurant and was organized by The Society of the Friends of the

People (Livio p. 130). During Alexandre Dumas’ toast, Evariste stood up holding a glass of wine and an open jackknife and shouted “To Louis-Philippe!” (Livio 130). A few days later Evariste was arrested and charged with threatening the king’s life; however

Galois was acquitted of any charges. In July of 1831, Evariste lead a 600 person protest march wearing his military uniform which was illegal by that time, carrying guns and knives (Livio p. 134). Evariste and his friend Ernest Duchatelet were arrested, went to trial and received prison time, however, Evariste’s sentence seemed to be longer than most, six months imprisonment, while Duchatelet received only three (Livio p. 136).

One night Evariste was talking to his friend Raspail, who was in prison for writing a letter that was found to be offensive to the King and national guard, and told him of his pain he still felt over his father’s death, then in a surprisingly accurate vision of his own death

Evariste told Raspail, “I will die in a duel on the occasion of some coquette of low class”

(Rigatelli p. 85). By letters recovered it has been determined that Evariste had a brief Shaffer 15 love affair with a woman named Stephanie Porterin du Motel, it is claimed that Galois fell intensely in love with this woman who wasn’t too crazy about him to begin with (Livio p. 140). When the affair was terminated by Stephanie, Evariste was devastated; it has been speculate that it was this relationship the eventually led to Galois’ death (Livio p.

141).

Evariste Galois’ death is shrouded in mystery, he wrote a series of letters the night before his death; the most amazing fact is that he worked out and defined the outline for what is now known as Galois theory. Evariste wrote a letter to his republican comrades, to two of his republican friends N.L. (most likely Neopolean Lebon), and V.D.

(most likely Vincent Delaunay), the last letter was to Galois’ friend Auguste Chevalier and this letter was the one that entailed his outlined theory (Livio p. 143-44/ Waerden p.

105). The theory was sketched on a piece of paper, letters were scribbled out and words rewritten; Evariste having made some last minute changes and corrections to what is now known as Galois theory the night before the duel made one final quote “Je n’ ai pas le temps” – “I have no time” (Livio p. 145). Most believe that Galois faced an old friend in a duel on the morning of May 30th. The autopsy report concluded that

Evariste died from a gunshot to the right side of his stomach. It is documented that

Evariste died on May 31st 1832 at the age of 20 years and seven months; the day after the duel took place, in the presence of family members (Waerden p. 103). Evariste

Galois’ death was not largely reported on in any of the newspapers (Gillispie p. 259).

Due to that fact there are many conspiracy theories with regards to Galois’ death.

Some believe that the real killers were either; some of Evariste’s political enemies, a Shaffer 16 prostitute, a secret agent, or even one of his acquaintances; his death still remains a mystery almost two hundred years later.

Evariste Galois brilliant life was over in less than 21 years; but let it never be said that Galois didn’t experience life; he experience a love of learning, he was expelled from his school for his beliefs, he was sent to prison for a political protest, he was blacklisted from society, his reputation was bashed in the newspapers, he had loved and lost, and through all of this he revolutionize modern algebra with his research that has developed into Galois theory. His death is still a mystery, why was Galois shot and who did was responsible. Evariste Galois has gone down in history as a genius mathematician wise beyond his years that revolutionized algebra and met a premature demise. Shaffer 17

References

Gallian, Joseph A. (2002). Contemporary Abstract Algebra Fifth ed. Boston, MA:

Houghton Mifflin Company

Gillispie, Charles C. (Ed.). (1972) Dictionary of Scientific Biography Vol. V. New York,

NY: Charles Scribner’s Sons

Grattan-Guiness (Eds.). (1994). Companion Encyclopedia of the History and Philosophy

of the Mathematical Science. New York, NY: Routledge Inc

Infeld, L. (1948). Whom the Gods Love: The Story of Evariste Galois. New York, NY:

McGraw- Hill Book Company, Inc

Livio, M. (2005). The Equation That Couldn’t Be Solved: How Mathematics Genius

Discovered the Language of Symmetry. New York, NY: Simon & Schuster

Rigatelli, Laura T., translated by John Denton (1996). Evariste Galois 1811-1832: Vita

Mathematica Vol. 11. Boston, MA: Birkhauser Verlag

Waerden, van der, B. L. (1990). A History of Algebra: From al-Kahwarizmi to Emmy

Noether. New York, NY: Springer-Verlag

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