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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.
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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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The signature of a metric tensor is defined as the signature of the corresponding quadratic form. [2] It is the number (v, p, r) of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities.