In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood.
Originally Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.
Further abstraction of Galois theory is achieved by the theory of Galois connections.
Read more about Galois Theory: Application To Classical Problems, History, Permutation Group Approach To Galois Theory, Modern Approach By Field Theory, Solvable Groups and Solution By Radicals, Inverse Galois Problem
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... combinatorics, differential geometry, topology, partial differential equations, and Galois theory ... In Emil Artin's book Galois Theory, Milgram also discussed some applications of Galois theory ... Satzes von Rédei (a result in graph theory) with Tibor Gallai in 1960 ...
... Liouville's theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true ... The theorem can be proved without any use of Galois theory ... Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the constants (correspondi ...
... This yields the approach to solving polynomial equations in terms of inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomials in the roots), how can one recover the roots? This leads to studying solutions of polynomials in terms of the permutation group of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory. ...
... Galois has developed a powerful, fundamental algebraic theory in mathematics that provides very efficient computations for certain algebraic problems by utilizing the algebraic concept of groups, which ... and fully validated extensions of Galois' theory ... in the 1960s, the development of even more powerful extensions of the original Galois's theory for groups by utilizing categories, functors and natural transformations, as well as ...
... All finite groups do occur as Galois groups ... to construct field extensions with any given finite group as Galois group, as long as one does not also specify the ground field ... The Galois group of F/L is S, by a basic result of Emil Artin ...
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