---
created: 2022-04-05T23:42
updated: 2024-03-04T19:00
aliases:
  - ramification index
  - ramification
  - ramified
  - inertia
  - inertia group
  - inertia degree
  - split
  - split prime
  - splitting
  - inert
  - inert prime
  - ramified prime
  - decomposition group
  - wild inertia
  - tame inertia
  - unramified
---

---

- Tags: 
	- #todo/too-long
- Refs:
	- #todo/add-references
- Links:
	- [discriminants](Unsorted/discriminants.md)
	- [different](Unsorted/different.md)
	- [extension](Unsorted/contraction%20(ideals).md)

---

![](2024-03-04.png)
# ramification index

Let $K$ be a local field with residue characteristic $p$, 

- $I=\operatorname{Gal}\left(K^{\text {sep }} / K^{n r}\right)$ its inertia group, 
- $I_p=$ $\operatorname{Gal}\left(K^{\text {sep }} / K^t\right)$ its wild inertia group, and 
- $I_t=I / I_p=\operatorname{Gal}\left(K^t / K^{n r}\right)$ its tame inertia group.

![](attachments/2023-03-05unram.png)

# 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**:

\begin{tikzcd}
	{} && L && {\OO_L = \intcl_L(\OO_K)} && {\mfp^e = \mfp \OO_L = \prod \mfp_i^{e_i}} \\
	\\
	{} && K && {\OO_K} && \mfp \\
	\\
	&& {}
	\arrow[hook, from=3-3, to=1-3]
	\arrow[hook', from=3-5, to=3-3]
	\arrow[hook, from=3-5, to=1-5]
	\arrow[hook', from=1-5, to=1-3]
	\arrow[shorten <=6pt, shorten >=6pt, squiggly, from=3-7, to=1-7]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=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)

# Motivation

Topological:
![](attachments/Pasted%20image%2020220213213955.png)
![](attachments/Pasted%20image%2020220213214217.png)
![](attachments/Pasted%20image%2020220213214305.png)

# ramification index

## In number theory

![](attachments/Pasted%20image%2020220124092205.png)
![](attachments/Pasted%20image%2020220124092245.png)
![](attachments/Pasted%20image%2020220124092432.png)
![](attachments/Pasted%20image%2020220126220602.png)

## In algebraic geometry

![](attachments/Pasted%20image%2020220214091557.png)
# Inertia Group

The [inertia group](https://www.wikiwand.com/en/Inertia_group) measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.

For $L\slice k$ a field extension where $k$ has a valuation $v$, write $S_v$ for equivalence classes on valuations of $L$ extending $v$.
Then $S_v \in \mods{G}$ for $G \da \Gal(L\slice k)$.
For $w$ lying over $v$, define

- The decomposition group of $w$ as $D_w \da \Stab_G(w)$.
- The valuation ring $R_w$ with maximal ideal $\mfm_w$.
- The inertian group of $w$ as $I_w \da \ts{\sigma \in G \st \sigma.x = x \mod \mfm_w\, \forall x\in R_w}$,

![](attachments/Pasted%20image%2020220318210307.png)

# Splitting Primes
![](attachments/Pasted%20image%2020220124093005.png)
![](attachments/Pasted image 20210514225852.png)

![Pasted image 20211105232939.png](Pasted%20image%2020211105232939.png)

# AG Interpretation
![](attachments/Pasted%20image%2020220124094952.png)

# Decomposition Group
![](attachments/Pasted%20image%2020220124095520.png)
![](attachments/Pasted%20image%2020220124095556.png)

![](attachments/Pasted%20image%2020220126215959.png)
See also [Artin symbol](Artin%20symbol.md)

![](attachments/Pasted%20image%2020220127134207.png)

# Tame and Wild Ramification
![](attachments/Pasted%20image%2020220124225018.png)

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](etale.md) extension $B / A$ of [Dedekind domains](Unsorted/Dedekind%20domain.md) is tame if and only if the [trace](Unsorted/field%20norm.md) $\operatorname{Tr}: B \rightarrow A$ is surjective.
# Results

![](attachments/Pasted%20image%2020220124224806.png)

# Examples
![](attachments/Pasted%20image%2020220124224955.png)
![](attachments/Pasted%20image%2020220124225045.png)

In the [Gaussian integers](Gaussian%20integers)
![](attachments/Pasted%20image%2020220128224913.png)

# Misc

![](attachments/Pasted%20image%2020220417014019.png)
![](attachments/Pasted%20image%2020220417014027.png)


# Misc

![](attachments/Pasted%20image%2020220510164813.png)