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ramification index

Idea

Idea: for extensions of number fields, take a prime in the ring of integers of the base, extend, and look at its prime factorization. If any \(e_i \geq 2\), the prime ramifies:

Link to Diagram

Motivation

Topological:

ramification index

In number theory

In algebraic geometry

# Inertia Group

The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.

For $L_{/ {k}} $ a field extension where \(k\) has a valuation \(v\), write \(S_v\) for equivalence classes on valuations of \(L\) extending \(v\). Then $S_v \in {}_{G}{\mathsf{Mod}} $ for \(G \coloneqq{ \mathsf{Gal}} (L_{/ {k}} )\). For \(w\) lying over \(v\), define

• The decomposition group of \(w\) as \(D_w \coloneqq{\operatorname{Stab}}_G(w)\).
• The valuation ring \(R_w\) with maximal ideal \({\mathfrak{m}}_w\).
• The inertian group of \(w\) as \(I_w \coloneqq\left\{{\sigma \in G {~\mathrel{\Big\vert}~}\sigma.x = x \operatorname{mod}{\mathfrak{m}}_w\, \forall x\in R_w}\right\}\),

Splitting Primes

Pasted image 20211105232939.png

AG Interpretation

Decomposition Group

See also Artin symbol

Tame and Wild Ramification

The ramification is tame when the ramification indices \(e_{i}\) are all relatively prime to the residue characteristic \(p\) of \(p\), otherwise wild. A finite generically etale extension \(B / A\) of Dedekind domains is tame if and only if the trace \(\operatorname{Tr}: B \rightarrow A\) is surjective. # Results

Examples

In the Gaussian integers

Misc

Misc