---
date: 2022-01-24 10:01
modification date: Monday 24th January 2022 10:01:10
title: Artin symbol
aliases: ["Artin symbol", "Frobenius element", "Frobenius", "Artin map", "Artin reciprocity"]

---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #NT/algebraic 
- Refs:
	- #todo/add-references
- Links:
	- [Unsorted/ramification index](Unsorted/ramification%20index.md)
	- [congruence subgroup](congruence%20subgroup.md)
	- [arithmetic Frobenius](arithmetic%20Frobenius.md)
	- [Kronecker symbol](Kronecker%20symbol.md).
	- [class field theory](Unsorted/class%20field%20theory.md)


---


# Artin symbol
![](attachments/Pasted%20image%2020220124100104.png)
![](attachments/Pasted%20image%2020220126162507.png)
# Artin Map
![](attachments/Pasted%20image%2020220124100224.png)

![](attachments/Pasted%20image%2020220126162728.png)


## Example: $\QQpadic$

![](attachments/Pasted%20image%2020220408133239.png)

![](attachments/Pasted%20image%2020220408133357.png)
![](attachments/Pasted%20image%2020220408133424.png)


# Artin Reciprocity

Statement of Artin reciprocity: the Artin map is surjective.
![](attachments/Pasted%20image%2020220126164003.png)
![](attachments/Pasted%20image%2020220126164030.png)

![](attachments/Pasted%20image%2020220126170059.png)
![](attachments/Pasted%20image%2020220126214037.png)

# Global Artin Map

![](attachments/Pasted%20image%2020220127134539.png)

# Examples

![](attachments/Pasted%20image%2020220126232933.png)