Evariste Galois

Evariste Galois was born on October 25, 1811, in Bourg-la-Reine, a small town just south of Paris ^[1]. His father, Nicolas-Gabriel Galois, was a cultivated liberal who became mayor of the town during the Hundred Days in 1815, and his mother, Adelaide-Marie Demante, came from a family of jurists. She was an educated woman with a firm grounding in Latin and classical literature, and she taught her son at home until he was twelve ^[2].

In October 1823 the boy entered the Lycee Louis-le-Grand in Paris, a school with a reputation for both academic rigor and political turbulence. His early record there was respectable in classics, and he won a prize in his first years, but his performance grew uneven. Teachers described a pupil who was original and quick yet withdrawn, and one report complained that he was consumed by a passion for mathematics to the neglect of everything else ^[1].

That passion had a clear starting point. Around the age of fourteen Galois encountered Legendre's textbook on geometry, which he is said to have read with the speed of a novel, and he soon moved on to the original memoirs of Lagrange on the resolution of algebraic equations ^[3]. Rather than working through the standard school curriculum, he went straight to the research literature, an approach that fed his talent but left gaps that examiners would later punish.

Anyone asking who was Evariste Galois in the late 1820s would have found a teenager convinced of his own powers and repeatedly blocked by institutions. In 1828 he sat the entrance examination for the Ecole Polytechnique, the most prestigious scientific school in France, a year early and without the usual course of preparation. He failed ^[2]. He remained at Louis-le-Grand, where the mathematician Louis-Paul-Emile Richard recognized his gifts and encouraged him to publish. In April 1829, at seventeen, Galois printed his first paper, a result on continued fractions, in Gergonne's Annales de mathematiques ^[3].

The same period brought a run of setbacks that shaped the rest of his life. In May and June 1829 he submitted his first researches on the solvability of equations to the Academie des Sciences, where Augustin-Louis Cauchy was appointed referee. The manuscripts were never reported on, and their exact fate remains uncertain, though evidence suggests Cauchy advised the young author to revise and resubmit rather than simply losing the work ^[4]. In July 1829 his father, hounded by a clerical faction in Bourg-la-Reine that had circulated forged verses under his name, killed himself. Weeks later Galois failed the Polytechnique entrance examination a second time, a result that admirers later blamed on examiners unable to follow his compressed reasoning ^[1].

He settled for the Ecole Preparatoire, the institution soon renamed the Ecole Normale, entering in late 1829. In February 1830 he submitted a fresh memoir on equations to the Academie in competition for the Grand Prize in mathematics. The secretary Joseph Fourier took the manuscript home and died in May 1830 before reviewing it; the paper was never found, and the prize went jointly to Niels Henrik Abel, posthumously, and Carl Gustav Jacobi ^[4]. Galois did publish three papers in the Bulletin de Ferussac that year, including one on number theory that introduced what are now called Galois fields, the finite fields used throughout modern algebra and coding theory ^[3].

Among Evariste Galois achievements, the central one is a complete answer to a question that had stood open for centuries: when can a polynomial equation be solved by radicals, that is, by a formula built from the coefficients using arithmetic operations and root extractions? Formulas of this kind had been known for equations of degree two since antiquity and for degrees three and four since the Italian algebraists of the sixteenth century. Paolo Ruffini and then Abel had shown that no such general formula exists for degree five, but their arguments did not explain which particular equations remain solvable ^[5].

Galois found the explanation by attaching to each equation a new kind of object, the group of permutations of its roots that preserve all algebraic relations among them. He showed that an equation is solvable by radicals exactly when this group, now called the Galois group, has a specific layered structure, the property known today as solvability of the group ^[5]. The insight did more than close a classical problem. It shifted attention from formulas to structure, and the concept of a group, which Galois was the first to use in something close to its modern sense, became one of the organizing ideas of mathematics ^[6].

His methods reached beyond the solvability question. The finite fields he constructed in his 1830 paper on number theory are fundamental in algebra, cryptography, and the error-correcting codes that protect digital communication ^[3]. In his final letter he also sketched ideas about what he called the theory of ambiguity and about integrals of algebraic functions, hints that later mathematicians connected to the theory of Riemann surfaces and abelian integrals ^[4]. The correspondence he established between subgroups and subfields, taught to every algebra student as the fundamental theorem of Galois theory, remains a model of how a good abstraction can dissolve a hard concrete problem ^[6].

Recognition came slowly. In January 1831 he submitted a rewritten memoir at the invitation of the Academie, and this time the referees, Simeon-Denis Poisson and Sylvestre Lacroix, did report: they found the arguments neither sufficiently clear nor sufficiently developed to judge their rigor, and returned the paper with a suggestion to expand the exposition ^[4]. Poisson's verdict was not absurd given the compressed style of the manuscript, but it meant that the theory stayed unpublished during its author's lifetime.

Galois came of age in the July Revolution of 1830 and threw himself into republican politics. While Parisians fought on the barricades that July, the director of the Ecole Normale locked his students inside the school; Galois, furious, later attacked the director in a letter published in the Gazette des Ecoles, and in early January 1831 he was expelled ^[2]. Freed from the school, he joined the Society of Friends of the People, one of the most militant republican organizations, and enlisted in the artillery of the National Guard, a unit dominated by opponents of the new king Louis-Philippe ^[1].

His politics soon put him in court. At a republican banquet on May 9, 1831, Galois raised a glass and a knife with a toast that was taken as a threat to the king. Arrested the next day, he was tried in June and acquitted after his defense argued that the words "if he betrays" had been drowned out by the noise of the room ^[2]. Freedom lasted a month. On July 14, 1831, he was arrested again for wearing the uniform of the disbanded artillery unit and carrying weapons at a Bastille Day demonstration. This time he was convicted and sentenced to six months in the Sainte-Pelagie prison, where he remained, apart from the trial period, until March 1832 ^[4].

Prison did not stop the mathematics. In Sainte-Pelagie he drafted a preface for his collected work, bitter about the Academie and the schools that had turned him away, and continued to revise his memoir on equations ^[4]. When a cholera epidemic swept Paris in the spring of 1832, the younger prisoners were transferred, and Galois spent his final weeks on parole at a pension run by a man named Faultrier on the southern edge of the city ^[1].

The circumstances of Galois's death have attracted speculation for nearly two centuries, and the sober record is thinner than the legends. During his weeks at the pension he became involved with a young woman, identified from fragments of his papers as Stephanie-Felicie Poterin du Motel, the daughter of a physician attached to the establishment. The attachment ended badly, and surviving drafts of his last letters speak of a broken affair and of a quarrel he felt honor-bound to accept ^[7]. Whether the duel that followed was a genuine affair of honor, a provocation by political enemies, or something the participants themselves half understood is not settled by the evidence, though most modern historians favor a private dispute over the romantic conspiracy theories ^[7].

What is certain is the outcome. On the night of May 29, 1832, expecting to die, Galois wrote to his friend Auguste Chevalier a long letter summarizing his mathematical discoveries, correcting his memoir, and sketching unpublished results, with a request that the letter be shown to Jacobi or Gauss so that they might judge the importance of the theorems ^[4]. The document, hurried and annotated with marginal notes, is among the most famous letters in the history of science.

Early on May 30, in a field near the Glacière area south of Paris, Galois was shot in the abdomen. He was found hours later, reportedly by a passerby or peasant, and carried to the Cochin hospital, where he died of peritonitis on May 31, 1832, at the age of twenty ^[1]. His brother Alfred was at his side. He was buried in a common grave at the Montparnasse cemetery, and no trace of the exact site remains; a memorial in Bourg-la-Reine commemorates him instead ^[2].

The rescue of Galois's work took fourteen years. Auguste Chevalier and Alfred Galois copied the manuscripts and sent them to leading mathematicians, and in 1843 Joseph Liouville announced to the Academie that he had verified the results. Liouville published the principal memoir and the testamentary letter in his Journal de mathematiques pures et appliquees in 1846, finally putting the theory into circulation ^[5]. Serious study followed: Enrico Betti developed the arguments in the 1850s, and Camille Jordan's Traite des substitutions of 1870 built a systematic theory of groups on Galois's foundations ^[6].

From there the influence widened far past the original problem. Galois theory became the standard framework for understanding field extensions and symmetry in algebra, and it resolved the classical impossibility questions of Greek geometry, including the trisection of the angle and the duplication of the cube, as corollaries ^[6]. Group theory grew into a discipline in its own right, indispensable in number theory, geometry, and the physics of symmetry, while Galois fields underpin practical technologies from data encryption to the codes on compact discs and deep-space transmissions ^[3].

The life itself became part of mathematical culture. The story of a twenty-year-old writing down the future of algebra on the eve of a duel has inspired novels, plays, and repeated historical reexamination, some of it corrective: scholars such as Tony Rothman have dismantled the more theatrical versions, including the claim that all the mathematics was invented in a single night ^[7]. Any accurate Evariste Galois biography has to hold both truths together, the reckless young republican and the disciplined thinker whose ideas were years ahead of his referees. Among the basic Evariste Galois facts, one is perhaps the most striking: the collected mathematical writings that changed algebra run to only about sixty printed pages ^[4].