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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 ...
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.
The Weil group of a class formation with fundamental classes u E/F ∈ H 2 (E/F, A F) is a kind of modified Galois group, introduced by Weil (1951) and used in various formulations of class field theory, and in particular in the Langlands program. If E/F is a normal layer, then the Weil group U of E/F is the extension 1 → A F → U → E/F → 1
The inertia group of w is the subgroup I w of G w consisting of elements σ such that σx ≡ x (mod m w) for all x in R w. In other words, I w consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of G w. The reduced ramification index e(w/v) is independent of w and is denoted ...
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.
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]