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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.
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.
The failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of the so-called ideal class group. This group is always finite. This group is always finite. The ring of integers O K {\displaystyle {\mathcal {O}}_{K}} possesses unique factorization if and only if it is a principal ring or, equivalently ...
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The ramification defect or ramification deficiency d of a valuation of a field K is given by [L:K]=defg where e is the ramification index, f is the inertia degree, and g is the number of extensions of the valuation to a larger field L. The number d is a power p δ of the characteristic p, and sometimes δ rather than d is called the ...
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