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Another useful class of examples of Galois groups with finite abelian groups comes from finite fields. If q is a prime power, and if = ...
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.
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 ...
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, Galois theory is a branch of abstract algebra. At the most basic level, it uses permutation groups to describe how the various roots of a given polynomial equation are related to each other.
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.
The decomposition groups in this case are both the trivial group {1}; indeed the automorphism σ switches the two primes (2 + 3i) and (2 − 3i), so it cannot be in the decomposition group of either prime. The inertia group, being a subgroup of the decomposition group, is also the trivial group. There are two residue fields, one for each prime,
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.