EL 118 Final Project: The Tragic History of Evariste Galois

A study of the fabled life of a brilliant mathematician using discrete models of mathematical logic

Dylan Cashman

DISCLAIMER: The author of this work did not intend for this to be a factual biography. It is in no way to be used as factual evidence in the study of Galois' life. It also is not a formal treatise, in any way, of modern mathematics. It is simply an exploratory project.The author of this work did not intend for this to be a factual biography. It is in no way to be used as factual evidence in the study of Galois' life. It also is not a formal treatise, in any way, of modern mathematics. It is simply an exploratory project.

The biography is divided up into seven different versions of the same story. They range from the most abstract, Biography 7, to the most literal, Biography 1.

y = x^2

Let G be the Galois group. The element Evariste is generated by Mother and Father. Evariste holds all of the properties of an element in the Galois group. Let us choose Evariste so that he is intelligent, and brilliant at mathematical intuition. Then we can assume he will end up in a prominent school.

We now assume, without loss of generality, that Evariste attends the Lycee Louis-le-Grand, excelling until, one year, he is told that he must repeat a grade because of his slight age. At this point we establish an isomorphism from the elements of Evariste's consciousness to the elements of a mathematician's - thus Evariste had become a mathematician.

When a child excels in school, it is not necessarily a good thing. It is tough to be separated from peers, and it is even tougher to have a reputation to live up to. And the greatest curse, for a precocious child like Evariste, may be that it is difficult to have a higher perception of any subject you study than those around you. Evariste was younger, Evariste was smarter, Evariste was smaller, Evariste was sharper. He did fine, but did he succeed? (did he?) Eventually he was held back, but it was out of jealousy. We cannot loft him above the other students - it is against our rules! We shall say that he is not mature enough, or we will call into doubt his studies, but either way we won't let him reach so far so early. And so Evariste was made to repeat his fourth year. School became a cyclic subset of his experience, with a period equal to

Sometimes repetition is good; we may learn things through induction, or by continuous iteration of an action. One way is that we may notice new aspects, different views of the same shape or path. On just his second walk-through of the penultimate year of his school, Evariste discovered something rather startling to newcomers - the elegant world of mathematics. Evariste was taught by the brilliant Legendre, bit by the bug, and his life was changed forever. We must ask the question - if Evariste was never introduced to Legendre, if Evariste was not a victim of the injustice that kept him back a year, where would he have ended up? Would he have been a politician, like his father? Would he have gotten into classics, like his mother? Was he an exception to the function of life, a defined discontinuity, or was he a constant - no matter what we plug into his life, he was to be a mathematician. However, these aims are rhetorical at best - we must wait to see if any light is shed on the topic in the future.

The boy read and read and read and read. Eventually, he finished. We may be able to better understand the boy if we imagine he is reading a comic book, or a youthful science fiction novel, and then we believe that he is reading a mathematical text containing incredibly complex notions. Legendre, the 74-year-old brilliant, accomplished, established mathematical leader and Galois, the 15-year-old brilliant, fresh, green academic, held long, sometimes intense discussions by candlelight in the book Elements de Geometrie in the cold dormitory of the Lycee Louis-le-Grand, but Galois was finished after two days, and he needed more.

The first hurdle that Evariste faced in his mathematical career was the entrance exam at Polytechnique. The exam was intended for adults well-versed in the contemporary mathematical language of the time. Administered orally, it required prospective students to thoroughly describe their problem-solving processes, elaborating on the accepted methods to showcase their understanding. Evariste had had little formal mathematical training, and subsequently failed the entrance exam. The following year, Louis-le-Grand offered an intense mathematics course taught by the competent and intelligent Louis-Paul-Emile Richard, and Evariste elected to enroll again in the school that he recently rejected.

Richard was an incredible teacher, but his best attribute was perhaps his ability to recognize brilliance. He instantly noticed Galois as being far ahead of other students, and encouraged Galois to explore any intuitions he had in the material. Possessing rare humility, Richard based some of his lessons on solutions crafted by Galois. In this benevolent atmosphere, Galois had his first mathematical paper published. While this paper was not more than a trifle compared to the ground-breaking theory that was being introduced at the period, it was not representative of the work of Galois. Richard recognized that the boy's theories were revolutionary, and with his encouragement, Galois wrote two memoires, or extensive mathematical treatises, and submitted them to the French Academie des Sciences to be reviewed by the most prominent mathematicians in France, including Augustin Louis Cauchy and Joseph Fourier. These memoires contained absolutely incredible work - groundbreaking theory on Algebra that was to completely transform the study of the subject. Beyond solving certain problems put out by the mathematical community, the works introduced the idea of treating algebra in completely axiomatic, abstract manner, depending strictly on logic. It was perhaps the most significant movement in mathematics of a period that included discoveries that allowed us to accurately model forces such as heat and fluid movement. It is mind-numbing to imagine such work was written by a seventeen-year-old.

Tragically, the works effectively disappeared. They were submitted by the instructor, Richard, in the care of Cauchy, with the idea that his prominence would allow their quick recognition throughout the field. However, Cauchy was a very self-centered man; the research of others was of little concern of his. The two memoires were taken home and subsequently lost by Cauchy. He forgot about the work of Galois, and gave no response to the eager young mathematician. Galois waited for more than a year for any word, but soon deduced that his work had been for naught.

His father killed himself