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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 ...
This map, known as the Artin map, is a crucial ingredient of class field theory, which studies the finite abelian extensions of a given number field K. [1] In the geometric analogue, for complex manifolds or algebraic geometry over an algebraically closed field, the concepts of decomposition group and inertia group coincide.
In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups. In this article, a local field is non-archimedean and has finite residue field.
The conductor of an abelian extension L/K of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let θ : I m → Gal(L/K) be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m.
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The group U (1) is called the group of principal units, and any element of it is called a principal unit. The full unit group O × {\displaystyle {\mathcal {O}}^{\times }} is denoted U (0) . The higher unit groups form a decreasing filtration of the unit group
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 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]