---
date: 2021-10-19 22:43
modification date: Thursday 21st October 2021 14:00:02
title: class field theory
aliases: [class field theory, CFT, local class field theory, global class field theory, nonabelian class field theory]

---

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


---

- Tags 
	- #NT/algebraic
- Refs:
	- <https://ocw.mit.edu/courses/mathematics/18-786-number-theory-ii-class-field-theory-spring-2016/lecture-notes>
	- <https://www3.nd.edu/~ajorza/courses/2014s-m80220/notes/lecture26.pdf>
	- <http://math.uchicago.edu/~may/REU2014/REUPapers/Lingle.pdf>
	- Notes on absolute Galois groups: [attachments/Absolute Galois Groups.pdf](attachments/Absolute%20Galois%20Groups.pdf)
	- <https://pub.math.leidenuniv.nl/~smitbde/edu/lcft_2011/index.php?menu=schedule>
	- Juvitop 2021 notes: [attachments/Juvitop_CFT_Notes.pdf](attachments/Juvitop_CFT_Notes.pdf)
	- Brief summary of main statements: <http://www-math.mit.edu/~poonen/papers/cft.pdf> #resources/summaries 
	- [https://kskedlaya.org/cft/preface-1.html](https://kskedlaya.org/cft/preface-1.html)
		- [https://math.ucsd.edu/~kkedlaya/math204b/](https://math.ucsd.edu/~kkedlaya/math204b/)
- Links:
	- [Local class field theory](Local%20class%20field%20theory)
	- [Global class field theory](Global%20class%20field%20theory)
	- [Artin map](Artin%20map.md)
	- [ray class](Unsorted/ray%20class%20field.md)
	- [Artin reciprocity](Unsorted/Artin%20symbol.md)
	- [Unsorted/Dedekind-Kummer theorem](Unsorted/Dedekind-Kummer%20theorem.md)
	- [Lubin-Tate](Unsorted/Lubin-Tate%20theory.md)

---


# class field theory

# What is a class field?

See [class group](Unsorted/class%20group.md), [fractional ideal](Unsorted/fractional%20ideal.md), [Galois module](Galois%20module), [totally unramified extension](totally%20unramified%20extension), [Artin map](Artin%20map.md).
![](attachments/Pasted%20image%2020220126221053.png)
![](attachments/Pasted%20image%2020220126220950.png)
![](attachments/Pasted%20image%2020220126220840.png)

# Local class field theory

See [profinite completion](Unsorted/profinite%20completion.md).
![](attachments/Pasted%20image%2020220127124134.png)
![](attachments/Pasted%20image%2020220126232434.png)
![](attachments/Pasted%20image%2020220126233042.png)
![](attachments/Pasted%20image%2020220127124453.png)

# Global Class Field Theory
![](attachments/Pasted%20image%2020220127135445.png)
![](attachments/Pasted%20image%2020220127135619.png)
![](attachments/Pasted%20image%2020220127135633.png)


# Results

- There is a correspondence between [Hecke characters](Hecke%20characters) and representations of abelian Galois groups.
  
# Nonabelian class field theory

![2023-01-09-3](attachments/2023-01-09-3.png)

# Notes

![](attachments/Pasted%20image%2020220127123917.png)

![](attachments/Pasted%20image%2020220126170006.png)
![](attachments/Pasted%20image%2020220127135712.png)
- Class field theory is [Langlands](Unsorted/Langlands.md) for $\GL_1$.
- ![Pasted image 20211105234852.png](Pasted%20image%2020211105234852.png)
- ![Pasted image 20211105235549.png](Pasted%20image%2020211105235549.png)
- ![Pasted image 20211106001614.png](Pasted%20image%2020211106001614.png)
![](attachments/Pasted%20image%2020220126164103.png)

![](CFT%20Course)