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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 ...
The inverse Galois problem is to find a field extension with a given Galois group. As long as one does not also specify the ground field, the problem is not very difficult, and all finite groups do occur as Galois groups. For showing this, one may proceed as follows. Choose a field K and a finite group G.
The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a bijection between the intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galois theory.
The field E H is a normal extension of F (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H is a normal subgroup of Gal(E/F). In this case, the restriction of the elements of Gal(E/F) to E H induces an isomorphism between Gal(E H /F) and the quotient group Gal(E/F)/H.
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory , so named in honor of ...
Galois extension A normal, separable field extension. Galois group The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups. Kummer theory The Galois theory of taking nth roots, given enough roots ...
The finite field (or Galois field) GF(125) = GF(5 3) has degree 3 over its subfield GF(5). More generally, if p is a prime and n, m are positive integers with n dividing m, then [GF(p m):GF(p n)] = m/n. The field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendental ...
Let / be a finite Galois extension of nonarchimedean local fields with finite residue fields / and Galois group .Then the following are equivalent. (i) / is unramified. (ii) / is a field, where is the maximal ideal of .