Évariste Galois: The Revolutionary Genius Who Founded Group Theory
A brilliant mathematician whose life was cut short, but whose ideas reshaped algebra and modern mathematics
Évariste Galois (25 October 1811 – 30–31 May 1832) was a French mathematician and political activist whose brilliant but tragically short life laid the foundation for Galois theory and the modern concept of group theory.
By the age of 20, he had revolutionized algebra by formulating a method to determine when a polynomial equation can be solved by radicals — a breakthrough that continues to shape mathematics today.
Though his life ended in a mysterious duel at just 20 years old, his ideas have endured, becoming a cornerstone of algebra, number theory, and modern cryptography.
📜 Quick Facts & Overview
Full Name: Évariste Galois
Born: 25 October 1811, Bourg-la-Reine, France
Died: 30–31 May 1832, Paris, France (aged 20)
Known For: Founding Galois theory, pioneering group theory
Legacy: His work solved the centuries-old problem of determining solvability of polynomial equations, influencing modern algebra, coding theory, and cryptography
Remarkable Fact: Galois wrote down the essence of his mathematical discoveries in letters on the night before his fatal duel
👨👩👦 Family, Childhood & Early Influences
Évariste was born in the village of Bourg-la-Reine (near Paris) to Nicolas-Gabriel Galois and Adélaïde-Marie Demante.
Father: Nicolas-Gabriel Galois was a respected local politician, a passionate supporter of liberal republican ideals, and head of the village school. His political activism would later influence Évariste’s own revolutionary convictions.
Mother: Adélaïde-Marie Demante provided Évariste’s early education. She taught him Latin, Greek, and classical literature at home until he was about 12, shaping his love of language and his sharp, polemical style of writing.
📚 Early Intellectual Spark
As a schoolboy, Évariste encountered Adrien-Marie Legendre’s Éléments de Géométrie — a demanding text far beyond the usual curriculum. He mastered it with ease and soon plunged into the works of Joseph-Louis Lagrange, engaging directly with advanced mathematical arguments at an age when most of his peers were just beginning basic calculus.
👨🏫 Recognition of Talent
His unusual abilities did not go unnoticed. At the prestigious Lycée Louis-le-Grand, his teachers quickly realized they had a student whose intellect and intensity set him apart, even among Paris’s brightest.
🎓 Formal Schooling, Exams & Early Publications
At age 17, Évariste Galois first attempted the highly competitive entrance examination for the École Polytechnique, the premier scientific school in France.
❌ Setback at Polytechnique
Despite his brilliance, he failed the entrance exam twice (1828 and 1829), likely due to his disdain for routine procedures and his inability to conform to exam styles that rewarded methodical exposition over originality.🏫 École Préparatoire / École Normale
Instead, he enrolled at the École préparatoire (soon renamed the École Normale), an institution intended to train teachers. Here, the spirited mathematics teacher Louis Richard recognized Galois’s unusual genius and encouraged him to pursue original research.✍️ First Publications (1829–1830)
Galois began publishing short but striking papers:Work on continued fractions
Articles in the Annales de mathématiques
Contributions to the Bulletin des sciences mathématiques
By the age of 18, he was already drafting the ideas that would crystallize into his celebrated mémoire on the solvability of polynomial equations by radicals — the seed of what we now call Galois theory.
🔬 The Mathematical Breakthrough — What Galois Actually Did
Évariste Galois transformed a centuries-old algebraic puzzle — “When can the roots of a polynomial be expressed by radicals?” — into a structural question about symmetry.
Instead of searching for formulas, he studied the permutations of the roots that preserved the equations’ relationships, connecting algebraic solvability with group structure.
🌀 Key Innovations
Galois Group: To each polynomial, Galois associated a group of symmetries (today called the Galois group) describing how its roots can be permuted without breaking algebraic relations.
Solvability by Radicals: He proved that a polynomial is solvable by radicals if and only if its Galois group has a specific subgroup structure — what we now describe as a chain of subgroups with abelian factor groups (a solvable group).
Primitive Equations: He introduced the concept of primitive equations and tied their difficulty to properties of their permutation groups.
Finite Fields: Galois also anticipated the idea of finite fields, which he called “imaginaries”. Today, these are formalized as GF(pⁿ) and play a central role in coding theory and cryptography.
🌍 Legacy of the Leap
Although his notation and presentation differ from modern algebra, Galois’s insight was revolutionary:
👉 Algebraic solvability is not about formulas, but about group-theoretic structure.
This conceptual leap became the cornerstone of modern abstract algebra and number theory.
📝 Manuscripts, Submissions & Editorial Misadventures
Galois’s revolutionary work faced significant hurdles before it could reach the wider mathematical community.
📜 Submission to the Paris Academy of Sciences
In January 1831, he submitted his “Premier Mémoire” — detailing conditions for solvability of polynomial equations by radicals — to the Paris Academy of Sciences.
Referee: Augustin-Louis Cauchy was assigned to review the manuscript. He suggested revisions, but the paper went through multiple hands and administrative delays.
Misfortune & Misplacement: Complications included lost manuscripts, the deaths of key referees, and critical assessments by Poisson and Lacroix.
❌ Lack of Publication in His Lifetime
Due to these difficulties, Galois’s work was not published while he was alive. Many of his papers were:
Scattered among friends or colleagues
Rejected as “sketchy” or “insufficiently developed”
Temporarily misplaced
Modern historians attribute these challenges to a combination of:
Galois’s concise, highly abstract style
Editorial mismanagement
The political and academic turbulence of the time
📖 Posthumous Publication
The complete collection of his mathematical writings was finally edited and published by Joseph Liouville in the Journal de Mathématiques Pures et Appliquées in 1846, bringing Galois’s ideas to the attention of the broader mathematical community and securing his lasting legacy.
⚔️ Politics, Expulsion & Imprisonment — Life Outside Mathematics
The revolutionary climate of Paris in 1830 drew Évariste Galois deeply into political activism.
📰 Republican Engagement
During the July Revolution of 1830, Galois became an active supporter of the republican cause, opposing the monarchy and conservative authorities.
He wrote a sharp public letter criticizing school authorities, which led to his expulsion from the École préparatoire in January 1831.
🏹 National Guard & Public Confrontations
Galois joined the National Guard’s artillery unit, which aligned with the republican movement.
He attended republican dinners and frequently provoked authorities, including an infamous incident where he brandished a dagger at a banquet in 1831.
⛓️ Arrest & Imprisonment
His political actions led to arrest and trial, and he spent several months in Sainte-Pélagie prison during 1831–1832.
While imprisoned, he continued to revise and compile his manuscripts, ensuring that his groundbreaking mathematical work progressed despite his confinement.
💡 Intersection of Politics & Mathematics
Galois’s political fervor was inseparable from his mathematical life. The tumultuous social and political environment influenced both the reception of his work and his opportunities for scholarly recognition.
💌 The Last Night, the “Testament” & the Duel
The final chapter of Évariste Galois’s life is both tragic and legendary, blending mathematics, politics, and personal drama.
✉️ The “Lettre Testamentaire”
On the night of 29 May 1832, Galois wrote a letter to his friend Auguste Chevalier, now famously called his “lettre testamentaire.”
The letter summarized key mathematical ideas and included several manuscripts, ensuring that his discoveries would survive beyond his short life.
❓ Motives and Mystery of the Duel
The exact reason for the duel remains uncertain:
Some evidence points to a quarrel involving a young woman, often identified as Stéphanie-Félicie Poterin du Motel.
The identity of his opponent is unclear, with newspapers and memoirs providing conflicting accounts.
⚔️ The Fatal Duel
Galois was shot on 30 May 1832 and succumbed to his wounds the following day (31 May 1832).
His final letters and manuscripts were instrumental in preserving his mathematical legacy.
🌟 Historical Clarifications
While popular legend claims he wrote all his major work the night before the duel, historians emphasize that much of his mathematics had already been composed earlier.
The testament primarily organized and summarized his work, rather than creating it all in a single night.
📖 Publication After Death & Early Reception
After Galois’s untimely death, his groundbreaking work finally reached the mathematical community through careful editing and advocacy.
✍️ Liouville’s Role
Joseph Liouville edited and published Galois’s collected mathematical works in the Journal de Mathématiques Pures et Appliquées in 1846.
Earlier, Liouville had announced Galois’s results to the Paris Academy in the 1840s.
Initially, Liouville’s commentary did not fully capture the group-theoretic depth of Galois’s work, leaving much of its revolutionary nature underappreciated.
🔄 Subsequent Mathematical Reformulation
Over the latter half of the 19th century, mathematicians such as Serret, Camille Jordan, Dedekind, Kronecker, and later Emmy Noether and Emil Artin:
Clarified and expanded Galois’s ideas
Translated his terse manuscripts into the modern language of field theory, group theory, and algebra
This process transformed Galois’s initial manuscripts into the textbooks and frameworks used by algebraists today.
💡 Key Insight:
The journey from Galois’s concise, abstract notes to fully developed algebraic theory required decades of careful reinterpretation, making his eventual recognition a testament to both his brilliance and the perseverance of later mathematicians.
🧮 Concrete Mathematical Legacy & Why Students Still Learn Galois Today
Évariste Galois’s work remains central to modern mathematics, and students encounter his ideas in algebra and number theory courses for several key reasons:
🔹 Deciding Solvability
Galois provided the first structural criterion for determining whether a polynomial is solvable by radicals, transforming a centuries-old problem into a systematic framework.
🔹 Galois Groups and Field Theory
His approach established the correspondence between subgroups of a Galois group and intermediate fields, now known as the fundamental Galois correspondence, which underpins much of modern algebra.
🔹 Finite Fields and Applications
Galois’s concept of “imaginaries” anticipated finite fields (GF(pⁿ)), which are critical in coding theory, cryptography, and algebraic geometry.
🔹 Birth of Abstract Algebra
By studying algebraic systems through their symmetry groups, Galois effectively laid the foundation for modern group theory and field theory, creating a structural viewpoint that reshaped mathematics.
💡 In Summary:
Galois solved a 350-year-old problem — the general solvability of polynomial equations — not with formulas, but by establishing a structural, group-theoretic theory whose descendants form the backbone of modern algebra.
📚 Sources &Further Reading
🏛️ Primary & Authoritative Sources
For accurate historical and mathematical research, consult the following sources:
Évariste Galois — MacTutor: Detailed biography and archival notes. ([Maths History][1])
“Galois theory” — Encyclopaedia Britannica: Clear summary of the mathematics and its impact. ([Encyclopedia Britannica][2])
Œuvres mathématiques: Posthumous edition edited by Joseph Liouville in Journal de Mathématiques Pures et Appliquées, 1846. Available through EUDML/Numdam. ([eudml.org][5])
Peter M. Neumann (ed.), The Mathematical Writings of Évariste Galois: Modern edition with translations and commentary (European Mathematical Society, 2011). ([ems.press][6])
“The Last Mathematical Testament of Galois”: Scholarly analysis of the 29 May letter and related manuscripts. ([Indian Academy of Sciences][4])
📖 Further Reading (Accessible & Scholarly)
Paul Dupuy, La vie d’Évariste Galois: Classic biography including letters and archival documents.
Ian Stewart, Galois Theory: Readable introduction for advanced undergraduates, with historical notes.
Jean-Pierre Ehrhardt, research articles on Galois’s life and the Academy’s reception.
❓ Frequently Asked Questions (FAQs)
Q1 — Did Galois prove that every quintic polynomial is unsolvable by radicals?
A: He provided the structural criterion showing why a general quintic is not solvable by radicals, connecting it to the non-solvability of the associated Galois group.
Q2 — Was the duel really over a woman?
A: The duel’s motive is unclear. Contemporary evidence points to a possible affair (often linked to Stéphanie-Félicie Poterin du Motel), but sources disagree on the opponent and exact cause.
Q3 — Did Galois write all his major work the night before the duel?
A: No — while the 29 May letter summarized key ideas, most of his work had been drafted earlier. The dramatic “all-at-once” narrative is exaggerated.
Q4 — Where can I access Galois’s original manuscripts or translations?
A: Liouville’s 1846 edition reproduces key texts; Neumann’s 2011 EMS volume provides modern translations with commentary.
Q5 — How should a student approach learning Galois theory today?
A: Start with field extensions and group theory, then study the fundamental theorem of Galois theory and examples (quadratics, cyclotomic extensions, solvable polynomials). Stewart’s textbook and modern algebra texts are ideal.