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Another definition of the Galois group comes from the Galois group of a polynomial []. If there is a field K / F {\displaystyle K/F} such that f {\displaystyle f} factors as a product of linear polynomials
Galois theory implies that, since the polynomial is irreducible, the Galois group has at least four elements. For proving that the Galois group consists of these four permutations, it suffices thus to show that every element of the Galois group is determined by the image of A, which can be shown as follows.
The absolute Galois group of an algebraically closed field is trivial.; The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R, and its degree over R is [C:R] = 2.
Its Galois group over the base field is the quotient group / = {[], []}, where [g] denotes the coset of g modulo H; that is, its only non-trivial automorphism is the complex conjugation g.
In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields.The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module.
The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. [a] A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension. [2]
Most of differential Galois theory is analogous to algebraic Galois theory. The significant difference in the structure is that the Galois group in differential Galois theory is an algebraic group, whereas in algebraic Galois theory, it is a profinite group equipped with the Krull topology.
Given a field K and a finite group H, one may pose the following question (the so called inverse Galois problem). Is there a Galois extension F/K with Galois group isomorphic to H. The embedding problem is a generalization of this problem: Let L/K be a Galois extension with Galois group G and let f : H → G be an epimorphism.