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Concept
Galois theory
Summary
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand.
Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, nth roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals.
Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cube and trisecting the angle), and characterizing the regular polygons that are constructible (this characterization was previously given by Gauss, but all known proofs that this characterization is complete require Galois theory).
Galois' work was published by Joseph Liouville fourteen years after his death. The theory took longer to become popular among mathematicians and to be well understood.
Galois theory has been generalized to Galois connections and Grothendieck's Galois theory.
The birth and development of Galois theory was caused by the following question, which was one of the main open mathematical questions until the beginning of 19th century:
Does there exist a formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?
The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist.
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Ontological neighbourhood
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Related publications (34)
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Related concepts (47)
Abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.
Fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.
Évariste Galois
Évariste Galois (gælˈwɑː; evaʁist ɡalwa; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years. His work laid the foundations for Galois theory and group theory, two major branches of abstract algebra. He was a staunch republican and was heavily involved in the political turmoil that surrounded the French Revolution of 1830.