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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 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 ...
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.
For example, in algebraic number theory, one often does Galois theory using number fields, finite fields or local fields as the base field. It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses the absolute Galois group of Q , defined to be the Galois ...
The motivating example comes from Galois theory: suppose L/K is a field extension. Let A be the set of all subfields of L that contain K, ordered by inclusion ⊆. If E is such a subfield, write Gal(L/E) for the group of field automorphisms of L that hold E fixed. Let B be the set of subgroups of Gal(L/K), ordered by inclusion ⊆.
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 ...
in other words the splitting field of P, then L/K is an example of a purely inseparable field extension. It is of degree p, but has no automorphism fixing K, other than the identity, because T 1/p is the unique root of P. This shows directly that Galois theory must here break down. A field such that there are no such extensions is called perfect.