Évariste Galois

The life of Évariste Galois

Timeline 1811Born on 25 October in Bourg-la-Reine, near Paris. 1823Enters the Louis-le-Grand college in Paris. 1827Discovers mathematics and abandons everything else. 1828Fails the École Polytechnique entrance examination for the first time. 1829Publishes his first paper. Sends work to the Academy. His father commits suicide. Second failure at the Polytechnique. Enters the École Normale. 1830The paper entered for the Grand Prize is lost when Fourier dies. July Revolution. 1831Expelled from the École Normale. Poisson rejects his memoir. Arrested twice. 1832Dies on 31 May, aged 20, after being wounded in a duel. 1846Joseph Liouville publishes his works. ◀ Previous Follow the journey Next ▶ 1811–1823 Bourg-la-ReineHe is born in the town where his father will be mayor. His mother teaches him at home until he is 12. 1823–1829 Louis-le-GrandAt school he discovers mathematics at 15, “the frenzy of mathematics dominates him”. Having failed the Polytechnique twice, he enters the École Normale in 1829 and is expelled in 1831. 1831 Pont NeufOn 14 July he is arrested on the bridge, wearing the uniform of the disbanded artillery and armed. 1831–1832 Sainte-Pélagie prisonHe spends eight months in prison. In July 1831 he receives Poisson’s rejection. 1832 Cochin hospitalWounded in the duel of 30 May, he is taken to the hospital, where he dies on the morning of 31 May, aged 20. Enable JavaScript to see the map. The stages are listed below. Galois’s Paris, from school to prison and hospital, within a few kilometres. Map: © OpenStreetMap contributors

October 15, 2026 Read

Shuffling the roots: the group of an equation

To jump with both feet over the calculations; to group the operations, to classify them according to their difficulties and not according to their forms: such is, in my view, the mission of future geometers. — Évariste Galois, preface to a manuscript (1831) Sixty years before Galois, Lagrange had looked at expressions made from the roots of an equation and counted how many values they take when the roots change places. Galois went one step further. Instead of looking at the expressions, he looked at the swaps. And not at all of them: only at those the equation allows.

October 15, 2026 Read

What a group is: the six symmetries of the triangle

Cut an equilateral triangle out of paper, number its corners 1, 2 and 3 and place it over its outline drawn on the table. In how many ways can you pick the triangle up and put it back so that it covers exactly the same outline? There are six. You can do nothing; turn it by $120°$ or $240°$; or flip the triangle over about one of its three axes, each passing through a corner and the midpoint of the opposite side.

October 15, 2026 Read

When an equation can be solved by radicals

With the idea of the group of an equation, Galois answered the question left open after Abel: which equations can be solved by radicals? The answer does not depend on any trick. It depends only on the shape of the group. Each root cuts the group Solving an equation by radicals means adding roots, one at a time, to the numbers already known. At the start we know only the rational numbers, and the group of the equation is whole. Each time we add a root, such as $\sqrt3$ or $\sqrt[3]{5}$, new true relations appear, and fewer permutations respect them. The group shrinks. When the group becomes just the identity, all the roots of the equation are written with known numbers: the equation is solved.

October 15, 2026 Read

Galois imaginaries: finite fields

In June 1830, aged 18, Galois published a short paper, On the theory of numbers, in a Paris scientific bulletin. In it he invented new numbers, which he called imaginaries, for a world in which only finitely many numbers exist. For more than a century the paper looked like a curiosity. Today it is inside every mobile phone. Counting with remainders In clock arithmetic we count only with the remainders of division by a fixed number. With the number 5 there are only $0, 1, 2, 3, 4$, and the sums “wrap around”:

October 15, 2026 Read

Doubling the cube and trisecting the angle: the impossible constructions

The Greeks left three construction problems that nobody managed to solve with ruler and compass, the tools of Euclid’s Elements: Doubling the cube: constructing the edge of a cube with twice the volume of a given cube. Trisecting the angle: dividing any angle into three equal parts. Squaring the circle: constructing a square with the same area as a given circle. Legend has it that, during a plague, the oracle of Delos told the Athenians to double the altar of Apollo, which had the shape of a cube. They doubled the edge, the volume became eight times larger, and the plague went on. For two thousand years mathematicians tried. The answer only came in the nineteenth century, with the ideas of Gauss and Galois: it is impossible.

October 15, 2026 Read

The legacy of Évariste Galois

This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind. — Hermann Weyl, Symmetry (1952) Galois left some sixty pages of mathematics, much of it written in haste. It took half a century for them to be understood. When they were, they changed the very idea of what mathematics is: instead of studying numbers and figures, studying the transformations that leave them unchanged.

October 15, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Galois at 15, drawing by an unknown artist: source page · public domain. Galois drawn by his brother Alfred, published in 1848: source page · public domain. The letter to Auguste Chevalier, 29 May 1832: source page · public domain. The Louis-le-Grand college, Paris, photo by Celette: source page · CC BY-SA 4.0. Blog figures The figure of the six symmetries of the triangle is an original illustration. The table in the post was checked by a program that composes the permutations of the corners. The figure has no text and serves every language of the site.

October 15, 2026 Read