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This states that given a finite Galois extension / ... Another useful class of examples of Galois groups with finite abelian groups comes from finite fields.
The Galois group of f(x) modulo 2 is cyclic of order 6, because f(x) modulo 2 factors into polynomials of orders 2 and 3, (x 2 + x + 1)(x 3 + x 2 + 1). f(x) modulo 3 has no linear or quadratic factor, and hence is irreducible. Thus its modulo 3 Galois group contains an element of order 5.
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]
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 E / F {\displaystyle E/F} is finite, the Krull topology is the discrete topology.
For example, if L is a Galois extension of a number field K, the ring of integers O L of L is a Galois module over O K for the Galois group of L/K (see Hilbert–Speiser theorem). If K is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of K and its study leads to local class field theory.
For example, the Artin–Schreier theorem asserts that the only finite absolute Galois groups are either trivial or of order 2, that is only two isomorphism classes. Every projective profinite group can be realized as an absolute Galois group of a pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries ...
Hilbert's irreducibility theorem has numerous applications in number theory and algebra.For example: The inverse Galois problem, Hilbert's original motivation.The theorem almost immediately implies that if a finite group G can be realized as the Galois group of a Galois extension N of
This is an example of a solvable group, and indeed, the solutions to this differential equation are elementary functions (trigonometric functions in this case). The differential Galois group of the Airy equation, ″ =, over the complex numbers is the special linear group of degree two, SL(2,C). This group is not solvable, indicating that its ...