We have mentioned that Évariste Galois (and independently Abel) proved the general insolvability of quintic equations. Galois' life is possibly the most dramatic of any great mathematician and so is worth a mention here. Galois' life was a struggle against almost everyone and everything he encountered: polynomial equations, organised incompetence, political injustice, and deep questions of pure mathematics. By the time of his death at age 20, Galois had been tried and acquitted for threatening the king, armed himself for revolution in the Republican artillery, and — although nobody knew it for another decade — revolutionised mathematics. He died in a pistol duel.
Galois was born near Paris in 1811. Educated first by his mother, he entered a preparatory school at age 12 and, bored with school classes, started reading the works of great mathematicians. By age 15 he was reading original research memoirs. But it was not only mathematics that interested him: French society at the time was engaged in an ongoing and bitter struggle between the democratic ideals of the Republic and the conservative forces of the monarchy. Like his father, who was mayor of Bourg-La-Reine, Galois was profoundly opposed to tyranny.
Attempting the entrance exam to the top university, the École Polytechnique, a year early, Galois failed. It was to be the first of many episodes of mathematical incomprehension by his supposed superiors. Some teachers and lecturers saw his ability, but their word was not sufficient for admission. He enrolled in a private mathematics course instead.
By age 18, Galois was thinking deeply on the question of the solvability of polynomials. He submitted an article with some of his results on the topic. Some accounts suggest that the article was lost or thrown out; other accounts suggest he was asked to resubmit another version. In any case, the paper was never published and nothing became known of its results.
Around the same time, Galois' father committed suicide after a vicious political dispute turned personal and a political enemy published scurrilous material in his name. Shortly afterwards, Galois reattempted the Polytechnique entrance exam. The apocryphal story goes that he lost his temper and stormed out in disgust, hurling a blackboard eraser at the examiner; in any case he again failed. He enrolled at the École Normale instead.
In 1830 Galois again submitted his research, this time for the Grand Prize of the Academy of Sciences. The paper was received but the referee died before reading it; the paper again was lost. However, political developments overshadowed mathematics: the French parliament was dissolved by King Charles~X; anti-monarchists gained a majority in the ensuing elections; the king attempted to suppress the press and parliament; and the so-called July Revolution broke out. After a mass uprising, the royalists and the Republicans reached a compromise, and Louis-Philippe was made king. Galois desperately wanted to join the uprising but the Director of the École Normale locked students in. Galois wrote a letter condemning the Director and was promptly expelled. He joined the Artillery of the National Guard; the artillerymen were almost entirely against the monarchy. But the king dissolved the Artillery as a security threat. At a banquet with his Republican colleagues in May 1831, Galois made a 'toast' to Louis-Philippe while holding a dagger. He was arrested and tried for threatening the king, but acquitted; sources report the jury was moved by his youth. Shortly afterwards he received notice that his manuscript on the solvability of equations was rejected as 'incomprehensible'.
On 14 July (Bastille day) 1831, Galois led a demonstration wearing the (now banned) uniform of the Artillery and heavily armed. Arrested, convicted and imprisoned, he was able to work on mathematics in jail until he was eventually paroled in early 1832.
Galois' freedom was not to last long. He became involved in a brief and tumultuous love affair, which ended with his rejection. Shortly afterwards, for his advances towards the woman, he was challenged to a duel. There has been a great deal of speculation over possible political motives for the duel; what is certain is that he lost. On the eve of the duel, he wrote the famous letter quoted at the start of this module, in which he attempted to explain his work. It was pistols at 25 paces; Galois was shot in the stomach and died the next day.
Rejected from the universities, a soldier of the Republic, contemptuous of the authorities, and finally slain by a comrade over (in his words) 'an infamous coquette', Galois died as tempestuously and as misunderstood as he lived. It was not until 1843 that the great mathematician Liouville announced that he had found, 'among the papers of Évariste Galois … a solution, as precise as it is profound, of this beautiful problem: whether or not there exists a solution by radicals …'