Évariste Galois

Évariste Galois
	Évariste Galois, French mathematician
Tradition	Mathematics, Algebra, Group theory
Influenced by	Joseph-Louis Lagrange, Augustin-Louis Cauchy, Carl Friedrich Gauss
Lifespan	1811–1832
Notable ideas	Founder of group theory; Galois theory; foundations of modern algebra
Occupation	Mathematician
Influenced	Émile Picard, Camille Jordan, Modern algebra, Abstract algebra
Wikidata	Q7091

Évariste Galois (1811–1832) was a brilliant French mathematician best known for laying the foundations of group theory and unlocking the theory of equations. In a tragically short life—he died at age 20 in a duel—Galois developed powerful new ideas about algebraic equations that solved long-standing puzzles. His work provided a criterion for when a polynomial equation can be solved by radicals (using only roots and basic arithmetic) and essentially created the abstract notion of a group of symmetries. By associating each equation to a permutation group acting on its roots, Galois showed that an equation is solvable by radicals exactly when this group has a special solvable structure. This insight forged a new algebraic approach that matured into modern group and field theory.

Galois’s life combined passionate politics (as a young republican caught up in the 1830 July Revolution) with intense mathematics. His career was marked by dramatic failures (lost manuscripts, rejected submissions) and by revolutionary fervor. Yet after his death his classmates and later mathematicians recognized the depth of his contributions: Joseph Liouville rescued and published Galois’s notes in the 1840s, and later Camille Jordan, Émile Artin and others developed his ideas into a comprehensive theory. Galois’s mix of ground-breaking mathematics and political romance has made him a legendary figure, sometimes mythologized, in the history of science.

This article traces Galois’s life and work, explaining his major ideas in accessible terms. It covers his early education and mathematical awakening, the substance of his mathematical contributions (especially his group-theoretic approach to equations), his turbulent political activities, and the complex reception of his work after his death. We also discuss controversies and later influence, and Galois’s far-reaching legacy in modern algebra and beyond.

Early Life and Education

Évariste Galois was born on 25 October 1811 in Bourg-la-Reine, a suburb of Paris. His family was cultivated and politically engaged. His father, Nicolas-Gabriel Galois, was an educated man with republican sympathies and even became mayor of Bourg-la-Reine during the revolutionary Hundred Days of 1815. His mother, Adélaïde-Marie, was well-versed in literature and religion and personally taught Évariste the classics (he learned Greek and Latin) until he was twelve. Despite this background, no one in his family was known for mathematical talent, and Évariste’s early academic record was unremarkable until his teenage years.

In 1823 (age 12), Galois enrolled at the Lycée Louis-le-Grand in Paris. At first he struggled with general coursework and rhetoric (he was asked to repeat a year because of poor grades in literature). But in 1827, under the guidance of the teacher Louis Richard, Évariste discovered a passion for mathematics. He convented fast through Legendre’s Éléments de géométrie as if “reading a novel,” and by age 15 he was absorbing advanced treatises of Lagrange on algebra. His reports noted a restless energy: teachers called him original and headstrong, even “bizarre,” but acknowledged he showed exceptional ability in pure mathematics. A mathematics master remarked that Galois never did any schoolwork, instead spending all his time in “the highest realms of mathematics.”

Galois’s schooling coincided with political turmoil in France. King Louis XVIII and later Charles X faced public discontent after the fall of Napoleon. The July Revolution of 1830 would soon overthrow Charles X. In this charged atmosphere, Galois formed republican views much like his father’s. He also repeatedly attempted to enter the École Polytechnique (the leading French university for science and mathematics). He failed the entrance exam in 1828 and again in 1829. Some historians suggest his second failure was partly due to emotional stress: his father had just committed suicide in July 1829, under scandalous circumstances in their hometown, and Galois was devastated.

After failing Polytechnique, Galois entered the École préparatoire (later École Normale Supérieure) at Louis-le-Grand with the permission of his dismayed examiners. He took the required Baccalauréat exams at the end of 1829, earning his diploma despite making a poor impression in literature (one examiner complained that the student knew “absolutely nothing” in literature despite mathematical talent). Galois’s mathematics examiner noted: “He is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research.” This foreshadowed Galois’s career of brilliant but hard-to-decipher work.

Major Mathematical Works and Ideas

Galois’s contributions center on algebra, specifically the theory of equations and the new concept of a group. His key achievement was to tie the solvability of polynomial equations to the structure of a group of permutations of their roots. In modern terms, he established what is now known as Galois theory. We present the main ideas without excessive technical detail:

Equations and radical solutions. For centuries, mathematicians had formulas (involving addition, multiplication, and taking square, cube or higher roots) that solved quadratic, cubic and quartic equations. But attempts to find a similar formula for fifth-degree (quintic) or higher equations failed. By the early 1800s, Paolo Ruffini (1799) and Niels Abel (1824) proved the general quintic cannot be solved by radicals, meaning there is no general formula using only arithmetic and root extractions for all quintic equations. Galois went further: he found exactly which equations can be solved by radicals and which cannot.

Permutation of roots and the Galois group. Galois’s revolutionary idea was to study an equation by looking at how its solutions (roots) can be permuted. Each way of rearranging the roots is viewed as a permutation, and these permutations form a mathematical structure that Galois called a group. In today’s language, the Galois group of an equation is the set of all permutations of its roots that respect the algebraic relationships between the roots. (Another viewpoint: it is the group of automorphisms of the field obtained by adjoining the roots to the rationals.) For example, the quadratic equation \(x^2-2=0\) has roots \(\sqrt{2}\) and \(-\sqrt{2}\); its Galois group has two elements (the identity and the map switching the two roots). Each element of the group is a symmetry of the roots. Higher-degree equations generally have larger permutation groups.

Solvability by radicals ⇔ solvable group. Galois discovered that radicals solutions exist exactly when the Galois group has a special structure called solvable. Concretely, a group is solvable if one can break it down step by step through a series of normal subgroups whose successive quotients are groups of prime order (simple abelian pieces). Each step in the series corresponds to a radical extension in solving the equation. In plainer terms, solvable groups can ultimately be built up from simple cyclic groups (like adding successive prime-number-ordered “layers” of symmetry). The remarkable theorem Galois identified is: An equation can be solved by radicals if and only if its Galois group is a solvable group. If the group is not solvable, no formula by radicals exists.

This criterion explained Abel’s result as a special case: the full symmetric group \(S_5\) on five elements (the standard group for a general quintic) is not solvable, so the general quintic has no radical solution. But for certain quintics with smaller groups (like those whose group is a subgroup of a cyclic or solvable group), specific radicals solutions do exist. Thus Galois’s theory gives both a no-go condition and a “yes-go” classification.

Abstract group concept and its elements. Galois essentially invented the abstract concept of a group in this context. He used the French word groupe in a way very close to the modern definition: a set of permutations under composition. He introduced notions equivalent to ro concepts like subgroup, normal subgroup and cosets, even if he did not formalize them as later algebra did. Importantly, Galois observed that the decomposition of the group into left and right cosets (which we now recognize as testing normality of a subgroup) was crucial for solvability. In his last letter he even wrote about “proper decompositions” of a group (what we call a normal subgroup) and broke down many examples of such groups.

Finte (Galois) fields and linear groups. Another major idea in Galois’s writings is the notion of a finite field (a field with finitely many elements). Today we call such objects Galois fields (or GF\((p^n)\)). Galois considered fields formed by adjoining a root of an irreducible polynomial to a prime field and showed these still behave like fields under arithmetic. He constructed examples of groups tied to these fields. For instance, he studied the group of invertible \(2\times 2\) matrices over a finite prime field (denoted GL(2,p)), and he even identified some of these groups as simple groups (having no nontrivial normal subgroups). Notably, Galois found that certain projective linear groups over finite fields are simple (this was remarkably ahead of his time in group classification).

Elliptic and Abelian integrals. Besides equations, Galois also touched on analysis. He corresponded about elliptic functions and the more general Abelian integrals (integrals of algebraic differentials on curves). In notes from 1831, he classified Abelian integrals into three types, extending work of Abel and Jacobi. Though less famous than his algebraic work, this hints at his wide mathematical range. He saw deep connections between these functions and the algebraic structures he studied, anticipating later algebraic geometry.

Continued fractions. Galois’s very first published paper (1829) gave a neat result in number theory: a continued fraction for a quadratic irrationality is purely periodic if and only if the irrationality was in reduced form. He proved a theorem about when the continued fraction expansion of \(\sqrt{n}\) is palindromic. This work showed his precocious ability, though it is relatively elementary compared to his later group theory.

In summary, Galois’s mathematical output, all written up in just a few manuscripts, laid the groundwork for modern algebra. He provided the fundamental link between symmetries of equations and what formulas can solve them. Although his papers were terse and often hard to parse, they contained virtually all the central ideas: Galois groups, solvable groups, finite fields, linear groups, and the solvability criterion. His approach was highly structural and abstract, focusing on general principles that apply to whole families of equations rather than on specific formulas. In that sense, Galois did not so much give a recipe for solving a particular equation, but rather pointed out the deep reasons why such solutions exist or fail to exist.

Political Activism and Personal Life

While at the École Normale (late 1820s), Galois became increasingly involved in the radical politics of the time. He wrote articles in student newspapers attacking school authorities and praised republican causes. After the July Revolution of 1830 overthrew King Charles X, Galois was angered when the new king Louis-Philippe fell short of a republic. In December 1830 he composed a fiery defense of his classmates who had been locked inside school during the riots; this letter earned him expulsion from the École Normale. He soon joined the National Guard’s artillery corps, a republican militia, but this was shut down by royal decree shortly after.

Galois’s outspokenness led him to frequent confrontations with the authorities. He stormed barricades during the July insurrection and later disrupted a royalist banquet with toasts to republican ideals—actions that intermittently got him jailed. In mid-1831 he faced trial for threatening the king (after a dinner speech) but was surprisingly acquitted. However, on Bastille Day 1831 (14 July), he was arrested again for wearing the National Guard uniform and carrying weapons in Paris. He spent several months in the infamous Sainte-Pélagie prison. Even there, he responded to setbacks by trying to improve his mathematics (Poisson’s report on his 1831 memoir had criticized its clarity, prompting Galois to draft a fuller exposition). He also attempted suicide in prison, stabbing himself, partly out of despair at politics and mathematics.

Galois’s personal life took a romantic turn near the end. In 1832, during a cholera epidemic at Sainte-Pélagie, he met Stéphanie-Félicité Poterin du Motel, the daughter of the prison doctor. They exchanged letters after his release in April. The significance of this episode is debated, but many contemporary accounts connect it to what happened next. On 30 May 1832, Galois fought a pistol duel with an artillery officer named Perscheux d’Herbinville (perhaps the same person identified by Alexandre Dumas as his opponent) or possibly with another friend in disguise. The exact cause of the duel remains obscure. Some suggest Galois challenged another man alleged to have slighted Stéphanie, while others think Galois himself staged the duel in a way to force the conflict. One report quotes Galois calling himself “the victim of an infamous coquette and her two dupes.” In any case, Galois was mortally wounded in the duel and died of peritonitis on 31 May 1832, the day after being shot. He was only 20 years old. His funeral evolved into a republican demonstration and even riots, illustrating his status as a tender point in the revolutionary aftermath.

The night before the duel, convinced he might die, Galois famously wrote frenetic letters to friends. In one addressed to his fellow mathematicians (Auguste Chevalier and others), he summarized his mathematical theory and appended several manuscripts of work still in progress. He also wrote his famous line in a margin: "I have no time; I have no time" – a cry for more hours to “complete this demonstration.” These last writings, later called his mathematical testament, became a vital source of Galois’s ideas. (Notably, the romantic legend that he spent the entire night before the duel dictating the whole of group theory is considered exaggerated; in reality he outlined and reviewed what he had already discovered.) Shortly after dawn on 31 May, Galois passed away in the Cochin Hospital in Paris. According to tradition, his last words to his younger brother were: “Don’t weep, Alfred! I need all my courage to die at twenty.”

Influence and Reception

Galois’s ideas were far ahead of their time, and initially they gained little attention. After his death, his young team of supporters (including his brother and friend Chevalier) made copies of his papers and tried to interest prominent mathematicians like Gauss and Jacobi, but found no recognition. It was Joseph Liouville who finally unlocked Galois’s legacy. In 1843 Liouville announced to the French Academy of Sciences that he had absorbed Galois’s concise solution to the problem of solvability (published in the Journal de Mathématiques Pures et Appliquées in 1846). Liouville found the solution “as correct as it is deep” to deciding when a prime-degree equation is solvable by radicals. He published Galois’s collected papers in 1846, saving them from oblivion.

Over the next few decades, mathematicians gradually understood and expanded Galois’s game-changing insights. A key milestone was Camille Jordan’s Traité des substitutions et des équations algébriques (1870), which systematically developed Galois’s ideas on groups of permutations (substitutions) and their connection to equations. Jordan introduced the modern concept of a composition series of subgroups, proving what became known as the Jordan–Hölder theorem about uniqueness of composition factors. His treatise made Galois’s theory accessible and applicable; German mathematician Felix Klein praised it for its clarity.

In the late 19th and early 20th centuries, group theory and field theory blossomed as core branches of algebra, often citing Galois at their origin. Mathematicians like Dedekind and Kronecker developed field theory; Émile Artin reformulated Galois theory in the 1920s using more modern algebraic language (introducing the concept of an abstract Galois group of a field extension). Artin’s work gave Galois theory its full modern form and broadened its reach (for example, to infinite extensions and to number theory).

Today Galois’s influence is immense. His name appears in many fundamental concepts: Galois group, Galois field (another name for a finite field), Galois connection (in lattice theory), and Galois cohomology (in number theory), among others. The unsolvability of the general quintic is a famous consequence of his theory, but the impact goes far beyond polynomial equations. Galois theory underpins much of algebraic number theory, giving tools to study field extensions and symmetries of numbers; it also influences algebraic geometry and modern cryptography (many cryptographic systems rely on arithmetic in finite fields, Galois fields). Group theory itself, born with Galois, became central in mathematics and theoretical physics as the language of symmetry.

Galois’s legacy is also honored in everyday life. In France one finds streets named after him (e.g. Rue Évariste-Galois in Paris’s 20th arrondissement). There is a lunar crater “Galois” named in his honor. The AFLAC stock ticker "GALOIS"? This is a bit trivial, but mathematicians and educators frequently celebrate his birthday (25 October) as Galois Day, commemorating his contributions. He is often depicted as a tragic young genius whose passionate life story inspires books and articles; for example, Harold Edwards and others have written about Galois’s mathematics and story, and at gatherings of mathematicians one sometimes hears Galois quoted as epitomizing youthful brilliance.

Critiques and Debates

Because Galois’s life is intertwined with drama, many accounts have mixed fact with romantic fiction. Scholars caution that several aspects of Galois’s story are uncertain or mythologized. For example, the exact reason for the duel that killed him remains a matter of debate. Early biographies sometimes claimed sordid motives or chivalric reasons; more recent research reexamines the letters and suggests that it may have been a matter of honor involving Stéphanie du Motel, or even a political plot (some argue Galois may have seen martyrdom as politically useful). No definitive answer exists, and historians note how Galois himself called the woman involved an “infamous coquette,” indicating hurt and betrayal, but full context is lost.

Similarly, the lore that Galois wrote out his entire theory on the eve of death is likely exaggerated. Contemporary sources confirm he worked on manuscripts that night, but a closer look shows he mainly annotated and reproved ideas he had already developed. His friends collected his papers afterward, and mathematician Hermann Weyl later praised one of Galois’s final letters for its depth—saying it might be “the most substantial piece of writing in the whole literature of mankind” (by some interpretations). Yet biographers note that Galois’s notes were often fragmentary and difficult to parse, and that even his contemporaries like Poisson failed to understand them.

Mathematically, Galois’s work was not controversial in content, but there were debates about credit and completeness. Abel’s and Ruffini’s earlier contributions to solvability were recognized, and some early readers wondered how much Galois independently knew of Abel’s results. In fact Galois built on Abel’s proof of the general quintic’s insolvability while introducing the novel group perspective. Modern historians like Carlo Ehrhardt argue that popular accounts of Galois often gloss over the interaction of ideas among many mathematicians of the time. Over time mathematicians corrected earlier orphaning of Galois: texts by Edwards and others explained the logical development of Galois’s theory in detail, resolving misunderstandings in the early expositions.

In summary, critiques of Galois’s legacy tend to focus on the legends around his life rather than on his mathematics, which is celebrated. Scholars have critically examined how narratives about Galois were constructed in the 19th and 20th centuries. Work by Laura Toti Rigatelli, Joseph Rotman, and others distinguishes the real historical records from the “icon” built later. They emphasize that Galois’s true miracle was purely intellectual. Indeed, despite the sensational circumstances of his death, Galois’s standing in mathematics rests securely on the originality and importance of his ideas.

Legacy

Évariste Galois’s legacy in mathematics is foundational. The concepts he introduced are now standard in algebra courses worldwide. Every modern algebra textbook discusses Galois groups, finite fields, and the solvability criterion. His insights are taught to undergraduates and graduate students alike. Abstract algebra—indeed, much of modern pure mathematics—would be unrecognizable without the structure he launched.

Beyond theory, countless applications trace back to Galois. For example, finite fields (called Galois fields) are the basis of error-correcting codes and cryptographic systems. The notion of symmetry groups underlies much of physics (such as the classification of elementary particles via group representations). Even number theorists working on very advanced problems often invoke Galois’s ideas – Andrew Wiles’s proof of Fermat’s Last Theorem in the 1990s, for instance, used deep Galois representations linking elliptic curves to modular forms. In Galois theory’s language, this was a triumph of algebraic symmetries applied to age-old equations.

Galois is also a cultural icon of science. He has appeared as a character or subject in historical novels, plays and documentaries, symbolizing the genius who perished young. While such portrayals sometimes dramatize his politics or romance, they also keep interest alive. Mathematicians remember Galois for his sheer audacity of thought; physicist Hermann Weyl’s praise is often quoted by educators presenting group theory to new students. Historians of mathematics cite Galois when illustrating the shift from solving specific numeric problems to studying abstract structures.

In cities and institutions, Galois’s name endures. There are streets and schools named after him in France (the Rue Évariste-Galois in Paris, for example). The lunar crater Galois commemorates his celestial legacy. He appears on French postage stamps as one of the great national scientists. In mathematics circles, the phrase “Galois theory” is ubiquitous, and seemingly simple techniques like “adjoining a square root” are often introduced by referring to his ideas.

Critiques and Debates

Although Galois’s mathematics is universally respected, people have debated how to interpret him as a historical figure. One debate involves the accuracy of the legend surrounding his last night and his letters. Biographers point out that while Galois did summarize his work for Chevalier, he had been working on many of those ideas for months beforehand. The image of him furiously writing all night is romantic but oversimplified. Similarly, accounts differ on why Galois entered the duel. Some early narratives blamed a love affair or personal rivalry, while recent historians suggest he may have instigated it intentionally—to endanger himself in a way that would disgrace King Louis-Philippe or protest a court verdict. In the absence of definitive evidence, scholars present multiple perspectives, noting that even Galois’s own friends gave conflicting stories (one account quotes Galois calling himself “the victim of an infamous coquette and her two dupes”).

In terms of mathematical debates, one issue was how to credit Galois’s originality. Criticisms of Galois were rare, but initially his work was so novel that even great mathematicians like Poisson and Lacroix found his papers unintelligible. Some early 20th-century historians questioned whether Galois deserved all the emphasis, since many later algebraists added rigor and generality. However, the consensus is that Galois’s core concepts were unique. His approach contrasted sharply with those of predecessors: whereas Lagrange and Abel had focused on specific equations, Galois introduced sweeping new language. Modern accounts, such as those by Peter Neumann and Harold Edwards, highlight that Abel had done part of the job (showing quintic insolubility) but Galois provided the structural reason. Thus, credit is generally shared historically: Abel solved the immediate problem while Galois forged the theory behind it.

Scholarly debates also touch on how Galois became an “icon.” Some historians like Caroline Ehrhardt explore how 19th-century biographers and writers shaped his image. For example, Édouard-Marie Cauchy (one commentator) and later popularizers sometimes cast Galois as a martyr poet of mathematics, analogizing him to a Roméo of algebra. Eloquently written but less accurate books (even by Nobel laureates like Leopold Infeld) have dramatized episodes in Galois’s life. In response, rigorous biographies (such as those by Rigatelli and Dupuy) sift fact from fiction. These works point out errors in common tales (e.g. the mistaken identity of the duel opponent). In short, modern scholarship treats the legends of Galois much as it does heroic myths of any era: with care for evidence and an effort to understand the real person beneath the mythology.

Legacy and Selected Works

Évariste Galois left a surprisingly rich written legacy for someone so young. His mathematical texts were finally published after his death, most notably in the Journal de Mathématiques Pures et Appliquées in 1846, edited by Liouville. Later, his Œuvres (collected works) were issued in book form (1897). For reference, some of Galois’s key publications and manuscripts include:

Démonstration d’un théoreme sur les fractions continues périodiques (1829). Galois’s first published paper, proving a result on periodic continued fractions (a property of quadratic irrational numbers).

Mémoire sur les conditions de résolubilité des équations par radicaux (submitted 1829, revised 1830, published 1846). His major memoir on when polynomial equations are solvable by radicals. Versions of this appeared in a lost manuscript to Cauchy (1829), a submission to the Paris Academy lost with Fourier (1830), and finally a version introduced in Liouville’s 1846 publication.

Notes on Abelian integrals (1830). In this unpublished manuscript he analyzed elliptic and more general algebraic integrals, classifying them (this work is found in his collected papers).

Affirmation of group structure (1831). Although never formally published in his lifetime, Galois’s letters to Chevalier dated 29-30 May 1832 effectively summarize his ideas. They outline the connection between groups of permutations and solvability. These notes were printed by Liouville and later by others, becoming the definitive reference to “Galois theory.”

Collected Works (Œuvres) (1897). Edited in Zurich by Kowalewski and others, this includes his published and unpublished manuscripts, plus a commentary on his theory.

Galois died before many of his writings could reach formal publication. Much of what we know comes from these posthumous sources. A concise timeline of Galois’s life highlights the key dates:

1811 – Born in Bourg-la-Reine, near Paris.

1827 – Begins serious mathematics under Louis Richard; attempts École Polytechnique (fails).

1829 – Father commits suicide; second failed Polytechnique exam; passes Baccalauréat; enters École Normale; publishes first paper (continued fractions).

1830 – July Revolution; expelled from École Normale for political writing; joins National Guard; publishes three short papers (elliptic integrals) in Bulletin of Férussac; father’s death and unrest.

1831 – Organizes private algebra class; submits theories to Académie des Sciences (rejected as “not clear” by Poisson); runs afoul of government; arrested (acquitted, then freed), then imprisoned second time.

1832 – Meets Stéphanie Poterin du Motel; fights duel on 30 May; dies on 31 May in Paris (age 20). Liouville later recovers his work (1843–46).

Galois’s selected works and ideas (often consumed by later generations) include:

Galois Theory (1846): The collection “Mémoire sur les conditions de résolubilité des équations par radicaux,” establishing his criterion for solvability of equations.

Group theory foundations: The introduction of groups, subgroups, and normal subgroups via permutation groups.

Finite (Galois) fields: The concept and construction of finite fields of prime-power order.

Rectifying continued fractions: Results on the periodicity and palindromicity of continued fraction expansions for quadratic irrationals (1829).

Abelian integrals: Notes classifying elliptic functions and more general integrals (1830).

Each of these works, although brief, was revolutionary in its time. Modern algebraic textbooks carry forward his ideas in chapters on Galois theory (often including explicit proofs of solvability criteria) and field extensions. The terms “Galois group” and “Galois extension” are standard, and even introductory discussions of the impossibility of angle trisection or squaring the circle may invoke Galois’s methods, since those problems reduce to questions about solvability of certain polynomials.

Conclusion

Évariste Galois stands as one of the most remarkable figures in mathematics. In a few intense years, and under extraordinary personal and political drama, he invented the abstract language of groups and fields that now underlie much of modern algebra. His criterion for solving polynomial equations transformed a centuries-old problem into a question of symmetry. Though his life ended in a romantic duel at age 20, his mathematical legacy long outlived him. Galois’s name is commemorated not only in theory — in every student’s course on Galois theory — but also in awards, monuments and even a lunar crater. His story reminds us that revolutionary ideas can arise even in turbulent times, and that true innovation sometimes comes from the bold vision of a young mind.