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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 ...
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,
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.
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.
Inertia is the natural tendency of objects in motion to stay in motion and objects at rest to stay at rest, unless a force causes the velocity to change. It is one of the fundamental principles in classical physics, and described by Isaac Newton in his first law of motion (also known as The Principle of Inertia). [1]
The ramification is tame when the ramification indices are all relatively prime to the residue characteristic p of , otherwise wild. This condition is important in Galois module theory. A finite generically étale extension B / A {\displaystyle B/A} of Dedekind domains is tame if and only if the trace Tr : B → A {\displaystyle \operatorname ...
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]
When is ramified, and I is the inertia group which is a subgroup of G, a similar construction is applied, but to the subspace of V fixed (pointwise) by I. [ note 1 ] The Artin L-function L ( ρ , s ) {\displaystyle L(\rho ,s)} is then the infinite product over all prime ideals p {\displaystyle {\mathfrak {p}}} of these factors.