---
date: 2021-10-21 18:42
modification date: Thursday 21st October 2021 18:42:50
title: Global field
aliases: ["global field", "global fields", "local field"]

---

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

---

- Tags: 
	- #NT/algebraic #arithmetic-geometry 
- Refs:
	- #todo/add-references
- Links:
	- [absolute value](Unsorted/absolute%20value.md)
	- [valuation](Unsorted/Valuations.md)

---

# global field

**Motto**: number field or function field.



![](attachments/Pasted%20image%2020220510164552.png)

![](attachments/Pasted%20image%2020220129175344.png)

# function fields
- Function field: an extension $F\slice k$ where $[F: k(x)] < \infty$ for some $x$ transcendental over $k$.

# global fields

![](attachments/Pasted%20image%2020220124121754.png)
![](attachments/Pasted%20image%2020220126094743.png)
![](attachments/Pasted%20image%2020220126224953.png)

 Global fields satisfy a product formula: 
$$
x\in K\smz \implies \displaystyle\prod_{v\in \Places(K)} \abs{x}_v = 1
$$

![](attachments/Pasted%20image%2020220129172449.png)
# local fields
![](attachments/Pasted%20image%2020220124122029.png)
![](attachments/Pasted%20image%2020220126232321.png)

- Idea: can arise as the rings of germs of functions, i.e. the local rings on a scheme.
- Arise as the completions of global fields.
- Another defintion: a field complete wrt a topology induced by a [discrete valuation](discrete%20valuation) with a finite residue field.

- Classification of local fields: 
	- Every local field is the completion of a global field wrt an absolute value.
	- An [archimedean](archimedean) local field is either $\RR$ or $\CC$.
	- A [nonarchimedean](nonarchimedean.md) local field is a finite extension $L/K$ for $K=\QQpadic$ or $\FF_q\fls{t}$
- The completion of $\ff(K)$ with respect to an [absolute value](Unsorted/absolute%20value.md)or [valuation](valuation) for $K$ a global field is a locally compact field, and thus a local field. 
-
# Examples
![](attachments/Pasted%20image%2020220124122104.png)

- Global fields
	- $\QQ$
	- Algebraic number fields $K\slice \QQ$
	- $L\slice K$ finite extensions of $K = \FF_q\rff{t}$, i.e. function fields of an [algebraic curve](Unsorted/algebraic%20curve.md) over a finite field
- Local fields:
	- $\RR$ and $\QQpadic$ for $p$ all primes in $\ZZ$ are local.
	- $\FF_q\fls{t}$ formal Laurent series over a finite field.
	- The completion of a global field at a [valuation](Unsorted/Valuations.md) / [absolute value](Unsorted/absolute%20value.md).
	- Nonexample: $\CC\fls{t}$, since its residue field is $\CC\fps{t}/\gens{t} \cong \CC$ which is not finite.

![](attachments/Pasted%20image%2020220124225444.png)