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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 = ...
For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method. [7] Joseph Alfred Serret who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook Cours d'algèbre supérieure .
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 ...
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 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,
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.