Galois

Évariste Galois (1811-1832) was a French mathematician who made major contributions to algebra, number theory, and group theory during his short lifespan.

Évariste Galois

Évariste Galois

At the age of 12, Galois joined Louis-le-Grand school where he had his troubled times. He felt that the school building with its grilled windows and gates was more like a prison than a school. He found that the school teachers were often arrogant and intimidating, and failed to motivate him and other students. Rebelling against their manner, he refused to study or do his assignments. The teachers demoted him to lower grade thinking he was stupid. Though Galois loved learning, he struggled with the textbooks the school used, and thought they were very elementary. He wanted to read the original texts and papers written by great thinkers and not a childish interpretation of their discoveries. Galois once found a copy of the famous French mathematician Legendre’s work on geometry, and it is believed that he read the entire text from cover to cover like a good novel. Galois often performed mathematical problems mentally, and skipping important steps, wrote his solutions in chicken scratches. This was one reason for his contention with his teachers who expected all his arguments and calculations written methodically on paper. Nonetheless, he gained a reputation as an argumentative student.

After high school, Galois wanted to join L’ École Polytechnique, a prestigious French college, but he failed the college entrance examination twice. In his first attempt at the entrance test, the examiners wanted to see all of Galois’s arguments and steps in problems presented methodically but as usual Galois had done most of them in his head. In his second attempt, he had an altercation with the examiners during the oral part of his examination, and in an angry spur of the moment, he hurled an eraser at one of the examiners hitting him in the face. After his second chance of getting accepted at L’ École Polytechnique disappeared abruptly, he began working on some original mathematical ideas on his own. He had already mastered the works of Legendre and Lagrange. Recognizing that Galois was a genius, Professor Louis-Paul-Emile Richard encouraged him to collect and send his discoveries to the elite group of scholars L’Académie française which was founded in 1635 by Cardinal Richelieu who was the Chief Minister of King Louis XIII. Following Professor Richard’s advice, Galois submitted a paper about his discoveries to Augustin-Louis Cauchy, a famous mathematician and professor at L’ École Polytechnique. However, Cauchy lost Galois’s paper and never had a chance to read it. Perhaps he had dismissed the work as coming from an amateur. After failing to get admission to L’ École Polytechnique, Galois eventually entered the L’École Normale, a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.

Galois made an important discovery about which equations could be solved using algebra and which ones could not be. He proved in a novel way that there can be no general formula for solving quintic equations which was proved earlier in 1824 by Niels Henrik Abel. Galois Galois submitted his work to the Academy of Sciences for the Grand Prize in mathematics. Galois’ most significant achievement in mathematics was his development of Galois theory and his contribution to Group theory. He noted that the algebraic solution to a polynomial equation was 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 could find a series of subgroups of its Galois group. He laid the foundation for Finite Fields theory. Galois fields has found important practical applications in Coding theory and Cryptography. Error Control Codes have made high speed communication possible in noisy network channels by providing an ability, at the receiving end, to correct data bits within digital messages that are corrupted during transmission. Without such codes, communication messages would require frequent retransmissions.

Galois submitted his work several times to the Academy of Sciences in competition for the Grand Prize in Mathematics but every time it was either not published for reasons unknown or it was lost. Galois’s friends suspected that someone jealous of his outstanding work had deliberately “misplaced” his papers. With many disappointments, Galois became bitter and started distrusting all institutions. He was also expelled from L’École Normale for criticizing the school’s director. Galois returned to mathematics after his expulsion from L’École Normale, although he continued to spend time in political activities. Wanting to fight injustice, he joined the Republicans, a forbidden radical group. The group spoke out for justice and the freedom of press. He was arrested several times for political acts and later acquitted. Once when he was imprisoned for six months, he continued developing his mathematical ideas in the prison.

Galois was killed in a duel in 1832 with under rather mysterious circumstances. The true motives behind the duel are obscure. Regardless of the reasons behind the duel, Galois was so convinced of his impending death that he spent the whole of previous night writing a letter his friend Auguste Chevalier outlining his ideas, and attaching three manuscripts. He scribbled in the margins, “There are a few things to be completed in this proof,” or “I do not have time to finish this.” He was clearly frustrated at not having enough time to complete his work. In closing his letter, Galois wrote “Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l’importance des théorèmes. Après cela, il y aura, j’espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis” which translates to “Ask Jacobi or Gauss publicly to give their opinion, not as to the veracity, but as to the importance of these theorems. In the future, I hope, some people will find it useful to decipher all this mess.”