Évariste Galois (1811 - 1832) was a French mathematician who, while still in his teens, laid the foundations of group theory and of the branch of algebra now known as Galois theory. [1] His work established precise conditions under which a polynomial equation is solvable by radicals, resolving a question that had occupied mathematicians for centuries. [2] He was also a committed republican activist, and he died from wounds received in a duel at the age of twenty, with his most important ideas still unpublished. [3]
Early life and education
Galois was born on 25 October 1811 in Bourg-la-Reine, near Paris, into an educated and politically engaged family. [1] He entered the Lycée Louis-le-Grand in Paris, where his early performance was uneven until he encountered mathematics, which quickly absorbed him. He read advanced works by leading mathematicians directly rather than relying on textbooks, and by his mid-teens he was pursuing original research. [2] He twice failed the entrance examination to the École Polytechnique, and instead enrolled at the École Normale Supérieure. [1]
Mathematical work
Galois turned to the classical problem of determining when a polynomial equation can be solved using radicals, that is, by a formula built from the coefficients using addition, multiplication, and root extraction. [2] Rather than seeking such formulas directly, he studied the symmetries among the roots of an equation, associating to each equation a group of permutations now called its Galois group. He showed that the equation is solvable by radicals precisely when this group has a particular structure. [3] In doing so he introduced concepts, including the notion of a normal subgroup, that became central to modern algebra. His approach unified and explained earlier results, such as the impossibility of a general algebraic solution to equations of degree five or higher. [2]
Political activism and death
Galois came of age amid the political turmoil of the July Revolution of 1830, and he became an outspoken republican. [1] He was expelled from the École Normale Supérieure after criticizing its director, and he was later arrested and imprisoned for his activism. [3] In May 1832, shortly after his release, he was drawn into a duel whose exact circumstances remain unclear. He was mortally wounded and died in Paris on 31 May 1832, at the age of twenty. [1] According to a widely repeated account, he spent the night before the duel writing out his mathematical ideas.
Legacy
Galois submitted memoirs to the Academy of Sciences during his lifetime, but they were not understood or were lost, and his central results were published only years after his death, largely through the efforts of Joseph Liouville in 1846. [2] Once appreciated, his work transformed algebra: group theory and Galois theory became foundational tools across mathematics, with applications reaching into number theory, geometry, and beyond. [3] His brief and dramatic life has also made him one of the most storied figures in the history of science.
