Galois was born at Bourg-la-Reine, near Paris, during the rule of Napoleon. He entered the Collège Royale de Louis-le-Grand in Paris in 1823 and it was here that his precocious mathematical genius first emerged. He published several papers while still a student and at the age of about 16 embarked upon his noted work on algebraic equations. But his career was marred by lack of advancement, associated with political bitterness. Twice, in 1827 and 1829, he was rejected by the Paris Ecole Polytechnique, and three papers submitted to the Academy of Sciences were rejected or lost. In 1830 he entered the Ecole Normale Supérieure to train as a teacher. That year revolution in Paris caused the abdication of Charles X, who was succeeded by Louis Philippe. Galois – fiercely republican – was expelled for writing an antiroyalist newspaper letter. In 1831 he was arrested twice: once for a speech against the king and the second time for wearing an illegal uniform and carrying arms – for this he received six-months' imprisonment. In the spring of 1832 he died in a duel; the details are uncertain but it may have been provoked by political opponents.

Galois seems to have anticipated that he was to die, for the night before was spent desperately recording his mathematical ideas in a letter to his former schoolmaster, Auguste Chevalier. Here he outlined his work on elliptic integrals and set out a theory of the roots (solutions) of equations, in which he considered the properties of permutations of the roots. Admissible permutations – ones in which the roots obey the same relations after permutation – form what is now known as a Galois group, having properties that throw light on the solvability of the equations. The manuscripts were published in 1846 and his work recognized. With the equally tragic Norwegian, Niels Henrik Abel, he is regarded as the founder of modern group theory.