---
date: 2022-01-15 21:49
modification date: Wednesday 9th February 2022 09:22:40
title: "p-adic integers"
aliases: ["p-adic integers", "p-adic", "p-complete integers"]

---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- [Some useful p-adic formulas](attachments/p_adic_formulas.pdf)
	- [Witt vector](Archive/AWS2019/Witt%20vectors.md)
- Links:
	- #todo/create-links 

---

# p-adic

See [disambiguating completion and localization](disambiguating%20completion%20and%20localization.md)

# p-adic integers

- Idea: invert all primes *except* $p$, so allow $a/b$ where $p$ does not divide $b$, i.e. $v_p(a/b) \geq 0$.
  
- Start with $\QQ$ and take the *Cauchy* completion with respect to the p-adic absolute value to obtain $\QQpadic$.
	- Yields alternative metric space completions of $\QQ$ that are not $\RR$. 
- Take the ring of integers to define $\ZZpadic = \OO_{\QQpadic}$. 
	- Note that $\ZZpadic \injects \QQpadic$, $\ZZ\injects  \ZZpadic$, $\QQ\injects \QQpadic$ are all dense embeddings.
- Can be realized as an [inverse imit](Unsorted/limit%20(categorical).md#Limits) exhibiting it as an [adic completion](Unsorted/adic%20completion.md)
$$
\ZZ_p := \inverselim_{n} \,\ZZ/\gens{p^n} \subseteq \prod_n \ZZ/p^n\ZZ
$$

- One can take the algebraic closure to obtain $\algcl \QQpadic$ and extend $v_p$.
  The resulting space is not Cauchy-complete, so complete to obtain $\CCpadic$.
  The result is algebraically closed by not locally compact.
- Interesting properties of any ultrametric space $(X, d)$:
	- All triangles are isosceles: if $d(x,y) \neq d(y,z)$ then $d(x, z) = \max\ts{d(x,y), d(y,z)}$, so two sides must have the same length.
	- Each point is the center of the disc it's in: if $x\in \DD_r(p)$ then $\DD_r(x) = \DD_r(p)$.
	- Every disc is clopen.
	- Two discs are either equal or disjoint, and if disjoint then $d(\DD_r(x), \DD_s(y)) = d(x,y)$, i.e. the infimum is obtained by the distance between any two points in either disc.
	- The induced topology is zero-dimensional, i.e. there is a basis of clopen sets.
	- $X$ is totally disconnected.
	- $\ts{x_k}$ is Cauchy iff *consecutive* differences vanish, i.e. $d(x_k, x_{k+1}) \to 0$.
	- If $X$ is Cauchy-complete, $\sum a_k$ converges in $X$ iff $a_k\to 0$.
- See [[spherically complete]]. a stronger notion than Cauchy completeness, needed for $p\dash$adic functional analysis.

## In terms of geometry and valuations

See [formal disk](Unsorted/formal%20disk.md)

- Can be realized in terms of valuations; **the p-adic integers form a ball**:
$$
\ZZpadic = \ts{x\in \QQpadic \st v_p(x) \geq 0} = \ts{x\in \QQpadic \st \abs{x}_p \leq 1} = \BB_p
$$
	- One can recover $\ZZ = \Intersect_{p\in \spec \ZZ} \BB_p = \Intersect_{p\in \spec \ZZ} \ZZpadic$ as the intersection of all balls.
	  
- $\ZZpadic$ is a local ring; **the maximal ideal is an open interior of a ball**: 
$$\mfm_{\ZZpadic} = p\ZZpadic = \ts{x\in \QQpadic \st v_p(x) > 0}  = \ts{x\in \QQpadic \st \abs{x}_p < 1} = \BB_p\interior = \ts{{a\over b}\in \QQ \st a\in p\ZZ }
$$

- The **units form a sphere**:
$$
\ZZpadic\units = \SS^1_p = \ts{x\in \QQpadic \st v_p(x) = 0} = \ts{x\in \QQpadic \st \abs{x}_p = 1}
$$

# Facts

- There is a **residue field** $\ZZpadic/p^n\ZZpadic \cong \ZZ/p\ZZ \cong \FF_p$ canonically.
- $\ZZpadic$ is a local but not complete ring.
- $\QQ\intersect \ZZpadic = \ZZplocal \subseteq \QQpadic$ is the **localization at p**.
- $\QQpadic \cong \ZZpadic\invert{p}$ as algebras.

- Topological properties of $\QQpadic$:
	- Every two elements can be separated by ball of non-integer distance, and the norm only takes on integer values. Thus all balls are clopen and $\QQpadic$ is totally disconnected.
	- $\ZZpadic$ is compact and $\QQpadic$ is locally compact; $\ZZpadic$ is totally bounded.
	- $\ZZpadic$ is a profinite group.
- **Open question**: can characterize $\ZZpadic = \ZZplocal\complete{\gens{p}}$, i.e. first localize at the prime ideal $p$ and then complete at the maximal ideal $p$.

# Expansions and series representations

- Series representations: $a\in \ZZpadic \implies a = (a_n) = f_a(p) \da \sum_{i\geq 0} a_i x^i \mid_{x=p}$ where $0\leq a_i \leq p^{i-1}$ and $a_i \cong a_{i+1}\mod p^n$ are successive lifts of $a_0$ modulo higher powers of $p$.
- $p\dash$adic expansions: the series representation is redundant since $a_i$ constrains $a_{i+1}$ to only the $p$ possible values in $\ZZ/p^{i+1}$ that are congruent to $a_i \mod p^i$. So one can always write $a_{i+1} = a_i + p^i b_i$ for $b_i = {a_{i+1} - a_i \over p^n}$. The sequence $(b_i)$ with $b_0 = a_1$ and $b_i = {a_{i+1} - a_i \over p^n}$ is the **p-adic expansion** of $p$.

# Examples

- Redundant representations in $\ZZpadic$ for $p=7$:
 ![](attachments/Pasted%20image%2020220124121320.png)
 - The more canonical $p\dash$adic expansions:
![](attachments/Pasted%20image%2020220124121515.png)

# Lifts

It almost never happens that a variety in characteristic p can be lifted to characteristic zero *together with its Frobenius endomorphism*.