ramification index

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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: attachments/Pasted%20image%2020220213213955.png attachments/Pasted%20image%2020220213214217.png attachments/Pasted%20image%2020220213214305.png

ramification index

In number theory

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In algebraic geometry

attachments/Pasted%20image%2020220214091557.png # 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 \(I_w \coloneqq\left\{{\sigma \in G {~\mathrel{\Big\vert}~}\sigma.x = x \operatorname{mod}{\mathfrak{m}}_w\, \forall x\in R_w}\right\}\),

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Splitting Primes

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AG Interpretation

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Decomposition Group

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attachments/Pasted%20image%2020220126215959.png See also Artin symbol

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Tame and Wild Ramification

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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

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Examples

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In the Gaussian integers attachments/Pasted%20image%2020220128224913.png

Misc

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Misc

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