Knowra Évariste Galois Évariste Galois Évariste Galois was a French mathematician whose work connected polynomial equations with permutation groups, founding what is now called Galois theory.
Polynomial : An expression formed from variables and coefficients using addition, subtraction, multiplication, and nonnegative integer powers. Galois studied when polynomial equations can be solved by radicals.
Galois theory : A mathematical theory relating field extensions to groups of automorphisms that preserve the base field. It formalizes the connection Galois established between equations and permutation groups.
Joseph-Louis Lagrange : An Italian-born mathematician whose work on permutations and equation-solving influenced later algebra. His analysis of root permutations anticipated a central idea in Galois's approach.
Camille Jordan : A French mathematician who developed group theory and gave a systematic exposition of Galois theory. Jordan's work made the theory more accessible and influential in later nineteenth-century mathematics.
Abstract algebra : The branch of mathematics that studies algebraic structures such as groups, rings, and fields. Galois's work helped shift algebra from formulas toward the study of structures.
Field (mathematics) : A set with addition, subtraction, multiplication, and division by nonzero elements, satisfying familiar arithmetic laws. Fields provide the number systems in which polynomial roots and their symmetries are studied.
Galois group : The group of field automorphisms of a splitting field that fix the base field. Its structure reveals whether an equation's roots can be expressed by radicals.
Paolo Ruffini : An Italian mathematician who argued that the general quintic equation cannot be solved by radicals. Ruffini pursued the impossibility problem that Galois later treated through group structure.
Richard Dedekind : A German mathematician whose work helped establish modern algebraic number theory and abstract algebra. Dedekind advanced the abstract treatment of fields and groups that grew from Galois's ideas.
Group theory : The mathematical study of groups and their actions, structure, and representations. His use of permutations helped establish groups as tools for solving algebraic problems.
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