---
created: 2022-04-05T23:42
updated: 2024-03-04T19:05
aliases:
  - Kronecker-Weber
---
---

- Tags 
	- #NT/algebraic
- Refs:
	- #todo/add-references
- Links:
	- [inverse Galois problem](Unsorted/inverse%20Galois%20problem.md)
	- [Hilbert 12](Unsorted/Hilbert%2012.md)
	- [class field theory](Unsorted/class%20field%20theory.md)

---

# Kronecker-Weber theorem

![](2024-03-04-1.png)

Now known as the "explicit case [global class field theory](Unsorted/class%20field%20theory.md)".

[Kronecker-Weber](Unsorted/Kronecker-Weber%20theorem.md) yields $\Gal(\QQ(\zeta_n)/\QQ) = C_n\units$ so $\Gal(\QQ^\ab/\QQ) = \varprojlim C_n\units = \prod_p \ZZpadic\units$.
For other global fields $K$, one has $\Gal(K^\ab/\QQ)\approx \dcosetl{\GL_1 K}{\GL_1 \AA_K}$ -- for $K=\QQ$ one has $\GL_1\AA_\QQ = \AA_\QQ\units = \RR_{\gt 0}\times \prod_p \ZZpadic\units$. Class field theory says one-dimensional reps of $\Gal(K^\ab/K)$ biject with one-dimensional reps of $\GL_1 \AA_K$, 

![](attachments/Pasted%20image%2020220126161412.png)
![](attachments/Pasted%20image%2020220126161427.png)

![](attachments/Pasted%20image%2020220730135305.png)