---
date: 2022-01-23 18:42
modification date: Sunday 23rd January 2022 18:42:01
title: Archimedean place
aliases: ["places", "Archimedean place", "Archimedean valuation", "nonarchimedean absolute value", "nonarchimedean field", "uniformizer", "valuation", "discrete valuation", "ultrametric", "place", "infinite place"]

---

Tags: 
#todo #NT/algebraic 
Refs:
[DVR](Unsorted/DVR.md)

# Valuation 
Motivation:
![](attachments/Pasted%20image%2020220509002411.png)


## For divisors in schemes

See [divisor](Unsorted/divisor.md).
![](attachments/Pasted%20image%2020220830181052.png)
Order of vanishing:
![](attachments/Pasted%20image%2020220830181126.png)

# Valuations (Definitions)


- Definition of a **valuation**: 
	- Start with any group morphism $$v\units: \GG_a(K\units)\to \GG_a(\RR),$$
	so $v(x + y)  = v(x) + v(y)$, satisfying an ultrametric-type inequality
$$
v(x+y) \geq \min (v(x), v(y))
$$
- Extend to 
$$
v: \GG_a(K)\to \GG_a(\RR) \union\ts{\infty} \qquad v(x) \da \begin{cases}v\units(x) & x\neq 0 \\ \infty & x=0 \end{cases}
$$
so that $v(x) = \infty \iff x=0$.

- Can then define an associated [absolute value](Unsorted/absolute%20value.md) 
$$\abs{x}_v := \exp(v(x))$$ where $e$ can be replaced with $c$ another constant.


- See also [valuation ring](valuation%20ring.md), unit groups, and the integral subring at [ring of integers of a nonarchimedean field](Unsorted/ring%20of%20integers.md#Ring%20of%20integers%20of%20a%20nonarchimedean%20field).

- The $p\dash$adic valuation $v_p: \QQ\to \ZZ$ is defined using unique factorization in $\QQ\units$:
$$
v_p(x) = v_p\left( \pm 1 \prod_{p_i\in \spec \ZZ} p_i^{e_i} \right) \da e_i, \quad v_p(0) \da \infty
$$
with an associated [nonarchimedean absolute value](Unsorted/absolute%20value.md)
$$
\abs{x}_p \da C^{-v_p(x)}, \quad c \in (0, 1),\quad  \abs{0}_p \da p^{-\infty}\da0
$$

- A **place** is an equivalence class of valuations.

- The **value group** of $v$ is the image $v(K) \leq \RR$.

- $v$ is a **discrete valuation**  if $v(K) \cong \ZZ \leq \RR$. See also [DVR](Unsorted/DVR.md).

- If $x\in K\units$, use that $v(x\inv) = v(1) - v(x) = -v(x)$, so either $v(x)\geq 0$ or $v(x\inv)\geq 0$.
	This forces the unit group of $A \da \ts{v(x)\leq 1}$ to be $A\units = \ts{v(x) = 0}$. Use this to partition $A$:
	- Positive valuation: non-units in $A$
	- Zero valuation: units in $A$
	- Negative valuation: $x\not \in A$ but $x\inv \in A$



# Places
- For function fields, maximal ideals $p$ of some valuations rings $\OO$.
If $p = \gens{t} = t\OO$, then $t$ is a uniformizer.

- Equivalence classes of valuations.

![](attachments/Pasted%20image%2020220126093310.png)
![](attachments/Pasted%20image%2020220126093333.png)
![](attachments/Pasted%20image%2020220126093359.png)
![](attachments/Pasted%20image%2020220126110633.png)

## Infinite places

**Infinite places** are those not arising from a prime in $\OO_K$.

![](attachments/Pasted%20image%2020220211133915.png)

![](attachments/Pasted%20image%2020220316133354.png)

## Embeddings

- For $L/K$, real places of $K$ are embeddings $K\embeds \CC$ with image in $\RR$, complex places are embeddings which intersect $\CC\sm\RR$.
- A real place $v$ of $K$ *is unramified* in this situation if all of the extensions of $v$ to $L$ are again totally real, and is ramified otherwise.
Complex places are always unramified.
- Example: in the extension $\mathbb{Q}\left(\zeta_{p}\right) / \mathbb{Q}\left(\zeta_{p}+\zeta_{p}^{-1}\right)$, where $\zeta_{p}$ is a primitive $p$-th root of unity for some odd prime $p$, the base field is totally real (all its Archimedean places are real) while the top field is totally imaginary, so all real places ramify in the extension. 
# Uniformizers
- Any element $\pi$ for which $v(\pi) = 1$ is a [uniformizer](uniformizer), so $\pi \in v_p(\bd \DD)$.
![](attachments/Pasted%20image%2020220214091425.png)


![](attachments/Pasted%20image%2020220123185220.png)

# Norms

![](attachments/Pasted%20image%2020220126094524.png)
![](attachments/Pasted%20image%2020220126094541.png)

See [Adeles](Unsorted/Adeles.md).

# Degrees and rational places

![](attachments/Pasted%20image%2020220316133013.png)

# Results

- Ostrowski's Theorem:
![attachments/Pasted image 20210511104707.png](attachments/Pasted%20image%2020210511104707.png)
- Valuation rings are integrally closed:
![](attachments/Pasted%20image%2020220123190306.png)

![](attachments/Pasted%20image%2020220124120608.png)
![](attachments/Pasted%20image%2020220124120616.png)

See [adic completion](adic%20completion.md)