Galois Theory

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

Other articles related to "galois theory, theory, galois":

Arthur Milgram
... 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 (differential Algebra) - Relationship With Differential Galois Theory
... 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 (corresponding to the constant ...
Esquisse D'un Programme - Abstract of Grothendieck's Programme - Extensions of Galois's Theory For Groups: Galois Groupoids, Categories and Functors
... Galois has developed a powerful, fundamental algebraic theory in mathematics that provides very efficient computations for certain algebraic problems ... begin with, Alexander Grothendieck stated in his proposal "Thus, the group of Galois is realized as the automorphism group of a concrete, pro-finite group which respects certain structures that are essential ... shall focus only on the well-established and fully validated extensions of Galois' theory ...
Galois Theory - Inverse Galois Problem
... All finite groups do occur as Galois groups ... It is easy 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 ...
Symmetric Polynomial - Applications - Galois Theory
... permutation group of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory ...

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