Galois and group theory

Evariste Galois was a French mathematician who died on May 31, 1832, aged 20, a day after being wounded in a duel in Paris. The night before, he wrote letters summarizing his discoveries, including one to his friend Auguste Chevalier asking that his ideas be published and judged by the leading mathematicians of the day. His central insight was that every polynomial equation hides a pattern of symmetry among its roots, now called its Galois group, and that this group decides whether the equation can be solved by a formula using only arithmetic and root extraction.¹²

The question was centuries old. Italian mathematicians had found formulas for cubic and quartic equations in the 1500s, published in Cardano's Ars Magna, and many expected the quintic (degree five) to follow. Joseph-Louis Lagrange had studied how shuffling the roots affects such formulas, and in 1824 the Norwegian Niels Henrik Abel proved that no general quintic formula exists. Galois went further and explained why, and which individual equations can be solved. Gauss's Disquisitiones Arithmeticae, with its abstract treatment of numbers and roots of unity, had helped set the style of thought he built on.¹

Galois's manuscripts were lost or rejected by the Academy of Sciences, partly because reviewers found them hard to follow, and he spent months in prison for his republican politics. Joseph Liouville read the papers and published them in 1846, and Camille Jordan's 1870 treatise spread the theory across Europe. The idea of understanding a structure through its symmetries grew into group theory, which now runs through modern algebra, crystallography, particle physics and cryptography.¹²