Galois

Évariste Galois was a French mathematician whose early 19th-century work revolutionized the study of polynomial equations and laid the foundations of group theory and field theory. In his brief career, he developed a theory describing when a polynomial equation can be solved by radicals by examining the symmetries among the roots. These symmetries form what is now called the Galois group, a group of permutations of the roots that preserves the algebraic relations with the polynomial’s coefficients.

Galois showed that the solvability of a polynomial by radicals is governed by the structure of its Galois group: polynomials whose Galois groups are solvable can be solved by radicals, while those with more complex groups generally cannot. This insight connects field extensions and automorphisms with permutation groups, and it provides a criterion for solvability that was previously unavailable. His ideas anticipate major developments in algebra, including the formalization of group theory and the theory of field extensions.

Galois’s manuscripts were not fully understood during his lifetime, and they were published posthumously in the 1830s. The work influenced later mathematicians and became central to modern algebra, shaping how scholars approach equations, symmetry, and structure. Galois was born in Paris in 1811 and died in 1832 at the age of 20, reportedly in a duel. Despite his brief life, his methods and concepts remain foundational, and the term Galois theory continues to be a cornerstone of contemporary mathematics.