The concept of Galois groups is a fundamental part of abstract algebra, particularly within the realm of field theory. Named after the French mathematician Évariste Galois, these groups describe the symmetries of algebraic equations, providing insights into their solvability by radicals.
Évariste Galois, in his brief life (1811-1832), made profound contributions to mathematics. His work, which was published posthumously, introduced the notion that the solvability of polynomial equations could be determined by examining the symmetry of their roots. Galois groups are essentially the group of automorphisms of the field extension of the rational numbers generated by the roots of a polynomial. This idea was revolutionary as it connected group theory with the solvability of equations, laying the foundation for modern abstract algebra.
The study of Galois groups has several significant implications:
Here are some practical examples and applications: