Galois and Ramanujan — Tragic Geniuses

Tragic Geniuses — The Ultimate Two

The tragic-genius lineage in the previous article reaches its extreme in Évariste Galois (1811–1832, dead in a duel at 20) and Srinivasa Ramanujan (1887–1920, dead of tuberculosis at 32).

• Galois — French republican activist. On the eve of a duel, in a single night, wrote out the whole of group theory in a letter, and died the next day at 20
• Ramanujan — self-taught South Indian. He wrote to Hardy at Cambridge with 120 formulas, spent five years in England, returned home, and died at 32

Both became myths of mathematical history. What they left, in the awareness that “there is no time left,” transformed 20th-century mathematics.

Évariste Galois — Founder of Group Theory, Dead at 20

Évariste Galois (1811–1832) was a French mathematician who died in a duel at 20 years and 7 months. In a few years of activity, he founded group theory and gave a structural explanation of why polynomial equations of degree ≥ 5 are not solvable by radicals.

His work was published by Liouville in 1846, more than a decade after his death, and only from the late 19th century did “Galois theory” become the core of modern algebra, number theory, and algebraic geometry. His life is called “the most dramatic in mathematics.”

Galois’s 21 Years

Year	Event
25 Oct 1811	Born in Bourg-la-Reine, near Paris. Father Nicolas-Gabriel is a liberal town mayor
1823	Enters Lycée Louis-le-Grand in Paris at 12
c. 1827	At 16 meets serious mathematics. Reads Legendre’s Éléments de géométrie in two days, absorbs Lagrange
1828	Fails the École Polytechnique entrance exam
May 1829	Submits first paper to the Paris Academy (Cauchy as referee)
July 1829	Father Nicolas commits suicide after slander by local clergy and royalists
Aug 1829	Fails Polytechnique entrance a second time (reportedly throwing an eraser at the examiner). Enters École Normale Supérieure
July 1830	July Revolution — throws himself into republican activism
Dec 1830	Expelled from the École Normale for a newspaper article criticizing the king
May 1831	Arrested for an anti-government speech; six months’ pre-trial detention
June 1831	Second paper to the Academy (Poisson referees, later rejected as “incomprehensible”)
July 1831	Rearrested at a revolutionary parade in arms; nine months’ imprisonment
29 Apr 1832	Released; convalescing in Paris during a cholera epidemic
29 May 1832	Challenged to a duel of honor over Stéphanie-Félicie du Motel
30 May 1832	Duel at Gentilly; shot in the abdomen
31 May 1832, 10 a.m.	Dies at Hôpital Cochin, 20 years and 7 months

The Night Before — “I Have No More Time”

The night before the duel, Galois — resigned to dying the next day — stayed up writing a testament letter to Chevalier, organizing his mathematical ideas. In the margins he wrote, again and again, “Je n’ai pas le temps” — “I have no more time.”

“What I ask of the public is not to judge the correctness of the theorems, but their importance.”

That night’s letter is the primary source for the content of Galois theory. “A 20-year-old, in a single night, sets down the foundation of modern algebra” — the most dramatic night in the history of mathematics.

Last Words

To his brother Alfred:

“Don’t cry, Alfred, I need all my courage to die at 20.”

The Duel — What Really Happened?

The identity of the opponent and the circumstances are still contested:

• Internal republican strife — a police trap
• Love triangle — over Stéphanie
• Political assassination

Stéphanie is variously said to have been the opponent’s fiancée or entirely unrelated. Galois refers to his opponent as “my friend d’H.” — an identification never fully resolved.

The Heart of Galois Theory

For polynomial equations aₙxⁿ + … + a₀ = 0 of degree ≥ 5, Galois asked whether they were solvable in radicals (using only arithmetic operations and root extractions), and decided the question through the structure of the Galois group:

• To each equation, associate the permutation group of its roots (Galois group)
• “Solvable in radicals” ⇔ “the Galois group is solvable”
• The Galois group of the general equation of degree ≥ 5 is the symmetric group S_5, which is not solvable (since A_5 is simple)
• Therefore the general equation of degree ≥ 5 is not solvable in radicals

This is Galois’s central theorem.

Relation to Abel

The Norwegian Niels Abel (1802–1829) in the previous article proved in 1824 that “the general quintic is not solvable by radicals.” But Abel did not structurally explain why — his was an impossibility proof. Galois built the general theory of which equations are solvable, deciding by the group. He went beyond Abel.

The Invention of “Group”

Galois introduced the term “group” (groupe) into mathematics in 1830. For him, “group” meant a permutation group; later, Cayley, Jordan, and Klein would generalize it to abstract groups.

Since the 20th century, groups have played fundamental roles in physics (symmetry, Lie groups), chemistry (molecular symmetry), crystallography, cryptography, and particle physics.

Galois Fields — Foundation of Modern Cryptography

Galois showed the existence and uniqueness of the finite field GF(p^n) of p^n elements (1830). It underpins modern cryptography and coding theory:

• Channel codes (Reed–Solomon)
• Elliptic-curve cryptography
• AES, RSA, and other algorithms

“Group theory from a young man dead in a duel at 20” — supporting the security of the internet 200 years later.

A Chain of Unfortunate Submissions

Year	Event
May 1829	First paper to the Academy; Cauchy referees; manuscript lost (Cauchy exiled during the July Revolution)
Feb 1830	Second paper to Fourier; Fourier dies immediately after, manuscript lost
June 1831	Third paper; Poisson refereed and rejected as “incomprehensible”
29 May 1832 (night)	Writes his testament letter summarizing the theory the night before the duel
1846	Liouville edits and publishes in the Journal de mathématiques pures et appliquées — 14 years after his death
1870	Jordan’s Traité des substitutions systematizes Galois theory — nearly 40 years after his death

Galois’s theory did not achieve wide recognition until almost 40 years after his death — a “posthumous glory” even more delayed than Abel’s.

Politics and Mathematics — Two Revolutions

Galois was a republican, anti-monarchist activist. After the July Revolution (1830) he took up arms as a National Guard:

• May 1831 — Arrested for a speech against Louis-Philippe
• July 1831 — Rearrested at an armed parade
• Continued mathematical work in prison

Galois’s mathematics and Galois’s politics are inseparable. Mathematics was revolutionary to him — it overturned the theory of equations — and politics was revolutionary in the same sense. Both revolutions, mathematical and political, are concentrated in 20 years.

Srinivasa Ramanujan — The Self-Taught Mystic

Srinivasa Ramanujan (1887–1920) was born in Erode, Tamil Nadu, in southern India. With almost no formal mathematical education he discovered vast numbers of formulas independently, and was “discovered” by the world through his letter to G. H. Hardy at Cambridge.

He died of tuberculosis at 32, but the three “Notebooks” and the incomplete “Lost Notebook” contain thousands of formulas still under active study.

Starting Point — G. S. Carr’s Book

Ramanujan grew up in a poor Brahmin family. As a boy he encountered G. S. Carr’s Synopsis of Elementary Results in Pure and Applied Mathematics, a list of theorems without proofs. He devoted himself to proving them all.

Uninterested in any subject at university but mathematics, he lost his scholarship. Working as a clerk at the Madras Port Trust, he developed his own mathematics.

“Proving and extending, alone, an entire book of unproven theorems” — a mode of self-teaching without parallel.

1913 — The Letter to Hardy

In 1913, at age 26, Ramanujan sent G. H. Hardy (1877–1947), then at the center of English mathematics, a letter enclosing his formulas.

The first letter contained about 120 strange and beautiful formulas, including:

1 + 2 + 3 + 4 + … = −1/12

Formal identities such as this (relying on analytic continuation of the zeta function) appeared with no explanation. Hardy, sensing a talent comparable to Riemann or Euler, judged that “only a genius could have written these”. With Littlewood, he resolved to invite Ramanujan to England.

Cambridge, 1914–1919

Ramanujan spent 1914 to 1919 at Cambridge. With Hardy he established the asymptotic formula for partition numbers (the circle method) — one of the monuments of modern analytic number theory.

The Partition Function p(n)

For the number p(n) of ways to write a positive integer n as an unordered sum of positive integers, Hardy and Ramanujan derived:

p(n) ~ (1 / 4n√3) · exp(π√(2n/3))

Rademacher later refined this into a fully convergent series.

Ramanujan Congruences

Ramanujan found many arithmetic congruences for p(n), such as p(5n + 4) ≡ 0 (mod 5) — the Ramanujan congruences, still central objects in modern number theory.

The Taxi Number 1729

An anecdote from Hardy’s visit:

Hardy: “The taxi number 1729 is a dull number.” Ramanujan: “No — it is the smallest number expressible as the sum of two cubes in two different ways.”

1729 = 1³ + 12³ = 9³ + 10³

Since then, the smallest number expressible as a sum of two cubes in n ways has been called a “taxicab number” — a symbol of mathematical intuition.

“The Goddess Namagiri Writes on My Tongue”

Ramanujan was a devout Hindu whose family goddess was Namagiri. He said that “the Goddess Namagiri writes formulas on my tongue.”

“Discovery before proof,” “formal calculation fused with intuition” — a pre-19th-century style, revived in the 20th. For five years, Hardy’s rigorous proof-driven mathematics and Ramanujan’s intuitive discoveries coexisted in one office.

Mock Theta Functions — A “Message in a Bottle” from His Deathbed

A few months before dying, Ramanujan wrote to Hardy from Kumbakonam a final letter introducing what he called “mock theta functions” — a new class of functions.

Their nature stayed mysterious for decades. In the 2000s, Zwegers’s dissertation identified them as the holomorphic parts of harmonic Maass forms, and they became a central topic of modern automorphic forms. A message in a bottle thrown toward the future from his deathbed.

Fast-Convergent Series for π

Ramanujan found families of formulas converging to π extraordinarily fast:

1/π = (2√2 / 9801) Σ_{k=0}^∞ (4k)! (1103 + 26390k) / ((k!)⁴ · 396^(4k))

Each term gives about 8 correct digits of π. This is the ancestor of modern high-precision π computation (the Chudnovsky brothers’ formula) — the basis of the algorithms behind trillion-digit π computations today.

Vegetarianism and Tuberculosis

Ramanujan was a strict vegetarian who struggled with the Cambridge diet. It is believed to have accelerated the progression of his tuberculosis.

Fell ill in 1917, returned to India in 1919, died in 1920 at 32. “Cultural distance shortened a genius’s life” — the London fog and cold, unfamiliar food, and separation from family combined to seal a short life.

Hardy’s Verdict

Hardy once rated mathematicians on a 1–100 scale:

• Himself — 25
• Littlewood — 30
• Hilbert — 80
• Ramanujan — 100

Asked what his greatest achievement was, Hardy answered “discovering Ramanujan.” One of the world’s greatest mathematicians ranked “finding an untrained Indian” as his own most important contribution.

Modern Legacy

Galois → the center of modern mathematics

• Abstract algebra — groups, rings, fields, modules
• Algebraic number theory — Galois representations, Iwasawa theory, the Langlands program
• Algebraic geometry — schemes, étale cohomology
• Mathematical physics — symmetry, gauge theory, the standard model of particle physics
• Information theory — cryptography, coding, error correction

Wiles and Fermat’s Last Theorem

Andrew Wiles’s proof (1995) rests on the Taniyama–Shimura conjecture linking elliptic curves’ Galois representations to modular forms. Impossible without Galois theory.

Ramanujan → contemporary revival

• Mock theta functions — a central topic of automorphic forms in the 2000s
• Black hole entropy — mock theta functions appear in quantum gravity (Zagier and others)
• π computation — modern computer science
• AI / machine learning — the Ramanujan Machine project, revived in the 2020s as automated formula discovery

In Film and Fiction

• The Man Who Knew Infinity (2015) — Dev Patel as Ramanujan, Jeremy Irons as Hardy
• The Indian Clerk (David Leavitt) — Ramanujan seen from Hardy’s inner life

The duel of Galois is a staple of mathematical fiction.

From This Article to the Next

We have traced the tragic geniuses of the 19th century. The next article turns to early 20th-century foundations — Hilbert, Poincaré, and Gödel. Hilbert’s attempt to found mathematics on a complete axiomatic system with his 23 problems (1900), Gödel’s incompleteness theorems (1931) that shattered it, and Poincaré, the father of topology — 30 years in which mathematics leaps into abstraction.