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over the field , then the Galois group of the polynomial is defined as the Galois group of / where is minimal among all such fields. Structure of Galois groups ...
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.
This makes a profinite group (in fact every profinite group can be realised as the Galois group of a Galois extension, see for example [1]). Note that when / is finite, the Krull topology is the discrete topology.
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]
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 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.
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers. This problem, first posed in the early 19th century, [ 1 ] is unsolved.
Analogously, given a path-connected topological space X, there is an antitone Galois connection between subgroups of the fundamental group π 1 (X) and path-connected covering spaces of X. In particular, if X is semi-locally simply connected , then for every subgroup G of π 1 ( X ) , there is a covering space with G as its fundamental group.