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The inertia group of w is the subgroup I w of G w consisting of elements ... is then called the v-th ramification group in upper numbering. In other words, ...
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,
More precisely, they are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character ε on the submodule.
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.
In that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric. A sequence of ramification groups is defined, reifying (amongst other things) wild (non-tame) ramification. This goes beyond the geometric analogue.
Ramification is a purely local property, i.e., depends only on the completions around the primes p and q i. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.
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ramification group A ramification group is a group of automorphisms of a ring R fixing some given prime ideal p and acting trivially on R/p n for some integer n>1. (When n=1 it is called the inertia group.) rank 1. Another older name for the height of a prime ideal. 2.