Title: Theory / Unimportant / Math / Analysis / p-adic
Tags: counterExample examplesSwitchExample1 hasSeenConjDef toBinaryRelation defToAlgebra setMultiplication toAlgebra

Definition: the highest \(m\) such that \(p^m\) divides \(n\)
Page: 2
Chapter: 1
Variables: \(p\) prime
\(n\in\mathbb N\)
Notation: \(\text{ord}_p(n)\)
Definition2: \(\infty\) when \(n=0\)
Conjdef: Either

Definition: \(\text{ord}_p(n)-\text{ord}_p(m)\)
Page: 2
Chapter: 1
Variables: \(p\) prime
\(n,m\in\mathbb N\)
Notation: \(\text{ord}_p(n/m)\)
Definition2: the greatest \(k\in\mathbb Z\) such that \(\frac{n}{mp^k}\in\mathbb Z\)
Conjdef: Equivalently

Definition: \(\frac{1}{p^{\text{ord}_p(\frac nm)} }\) if \(n\ne 0\)
Page: 2
Chapter: 1
Variables: \(p\) prime
\(n,m\in\mathbb N\)
Notation: [$]\abs{\frac{n}{m}}_p[/$]
Kind: Prop
Definition2: \(0\) if \(n=0\)
Conjdef: Either
DefType: The norm

Name: A non-Archimedian norm
Definition: A norm such that [$]\norm{x+y}\le\max(\norm{x},\norm{y})[/$] for each \(x,y\in\mathbb Q\)
Page: 3
Chapter: 1
Section: 2
Example: [$]\defFun{\mathbb Q}{\mathbb Q}{q}{\abs{q}_p=p^{\text{ord}_p(q)}}[/$]

Name: A non-Archimedian metric
Definition: A metric such that \(d(x,y)\le \max(d(x,z),d(z,y))\) for each \(x,y,z\in\mathbb Q\)
Page: 3
Chapter: 1
Section: 2
Extra: ultra metric

Smaller0: non-Archimedean
Greater0: non-Archemedean
Page: 3
Chapter: 1
Section: 2
Variables: [$]\norm{\cdot}[/$] a norm
\(d\) the metric associated to [$]\norm{\cdot}[/$]
Prefix Greater: \(d\) is
Prefix Smaller: [$]\norm{\cdot}[/$] is

Name: An Archimedean norm/distance
Definition: A norm/distance which is not non-Archimedean
Page: 3
Chapter: 1
Section: 2
Variables: [$]\norm{\cdot}[/$] a norm
\(d\) the metric associated to [$]\norm{\cdot}[/$]
Example: Standard norm on complex, real, rationals

Page: 3
Definition: The relation on distances which admits the same Cauchy sequences
Chapter: 1
Section: 2
Name: Equivalent distance

Page: 3
Definition: The relation on norm whose induced distances are equivalent
Chapter: 1
Section: 2
Name: Equivalent norm
Context: Analysis
Definition2: [$]\abs{}^\alpha[/$] and [$]\abs{}^{\alpha'}[/$] for \(\alpha,\alpha'\in(0,1]\)
Example: \(x\mapsto\rho^{\text{ord}_p(x)}\) and \(x\mapsto\rho^{\prime\text{ord}_p(x)}\), for two reals \(\rho,\rho'\in(0,1)\) on \(\mathbb Q\)

Name: The trivial norm
Definition: The norm sending 0 to 0 and everything else to 1
Page: 3
Chapter: 1
Section: 2

Smaller0: [$]\norm{\cdot}[/$] is trivial
Greater0: [$]\norm{\cdot}[/$] is [$]\abs{}_p[/$]
Page: 3
Chapter: 1
Section: 2
Variables: [$]\norm{\cdot}[/$] a norm on \(\mathbb Q\)
Kind: Theo
Smaller to greater: or

Smaller0: [$]\norm{x-y}[/$]
Greater0: [$]\min(\norm{x},\norm{y})[/$]
Page: 5, 6
Chapter: 1
Section: 2
Variables: [$]\norm{\cdot}[/$] a non-Archimedean norm
\(x,y\) elements of the space
Kind: Theo
Smaller to greater: =
Assuming: [$]\norm{x}\ne\norm{y}[/$]
Context: non-Archimedean norm

Smaller0: non-Archimedean
Greater0: isocele
Page: 6
Chapter: 1
Section: 2
Variables: \(d\) a distance on \(X\)
\(x,y,z\in X\)
Kind: Theo
Prefix Greater: the triangle \(xyz\) is
Prefix Smaller: \(d\) is

Name: The isoceles triangle principle
Definition: [$]\norm{x\pm y}\le\max(\norm{x},\norm{y})[/$] with equality if [$]\norm{x}\ne\norm{y}[/$]
Page: 6
Chapter: 1
Section: 2
Variables: \(d\) a distance on \(X\)
\(x,y,z\in X\)
Kind: Theo
Intuition: Intuitively, if two rational numbers are divisible by different powers of \(p\), then their difference is divisible precisely by the lower, power of \(p\)
Assuming: \(d\) non Archimedean

Smaller0: \(r_0=r_1\)
Greater0: \(D(x_0,r_0)=D(x_1,r_1)\)
Page: 6
Chapter: 1
Section: 2
Variables: \(d\) a distance on \(X\)
\(x_0,x_1\in X\)
\(r_0,r_1>0\)
\(D(x_i,r_i^r)\) the open disc of radius \(r_i\) and center \(x_i\)
Kind: Theo
Smaller to greater: iff
Assuming: \(d\) non Archimedean
Smaller1: \(d(x_0,x_1)<r\)
Extra: duplicate

Smaller0: \(r_0=r_1\)
Greater0: \(D(x_0,r_0)=D(x_1,r_1)\)
Page: 6
Chapter: 1
Section: 2
Variables: \(d\) a distance on \(X\)
\(x_0,x_1\in X\)
\(r_0,r_1>0\)
\(D(x_i,r_i)\) the closed disc of radius \(r_i\) and center \(x_i\)
Kind: Theo
Smaller to greater: iff
Assuming: \(d\) non Archimedean
Smaller1: \(d(x_0,x_1)\le r_1\)
Extra: duplicate

Name: Equivalent Cauchy sequences
Definition: Cauchy sequences \(\{a_i^1\}\) and \(\{a_i^2\}\) such that [$]\norm{a_i^1-a_i^2}\to0[/$] as \(i\to\infty\).
Page: 10
Chapter: 1
Section: 4
Variables: [$]\norm{\cdot}[/$] a norm on \(X\)

Definition: set is equivalence classes of Cauchy sequences for the norm [$]\abs{}_p[/$]
Page: 10
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
Notation: \(\mathbb Q_p\)
Definition2: with addition and product defined point-wise
Conjdef: The field
Definition3: inverse defined pointwise on a representant without 0

Definition: The constant Cauchy sequence \(x\)
Page: 10
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(x\in\mathbb Q\)
Notation: \(\{x\}\)

Definition: [$]\lim_{n\to\infty}\abs{a_n}_p[/$]
Page: 10
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(\{a_n\}\in\mathbb Q_p\)
Notation: [$]\abs{\{a_n\}}[/$]
Typ: A norm, \(\mathbb Q_p\to\{0\}\cup\{p^n\mid n\in\mathbb Z\}\)

Smaller0: [$]\abs{x}_p[/$]
Greater0: [$]\abs{\{x\}}_p[/$]
Page: 11
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(x\in\mathbb Q\)
Smaller to greater: =

Name: The \(p\)-adic expansion of \(x\)
Definition: [$]\{\frac{a'_i}{\abs{x}_p}\}[/$] such that \(a'_i\) is the canonical representation of [$]x\abs{x}_p[/$]
Page: 13
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(x\in\mathbb Q_p\)
Definition2: \(\sum_{i=-\text{ord}_p(x)}^\infty b_ip^i\)

Name: The \(p\)-adic expansion of \(x\)
Definition: \(0\le a_i<p^i\)
Page: 11, 13
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(x\in\mathbb Q_p\)
Context: \(p\)-adic
Definition2: \(a_i\equiv a_{i+1}\mod p^i\)
Conjdef: A Cauchy sequence \(\{a_i\}\) such that for \(i\ge1\)
Denoted by: \(\sum_{i=0}^\infty b_ip^i\)
Assuming: [$]\abs{x}\le1[/$]

Smaller0: [$]\abs{x}_p\le 1[/$]
Greater0: [$]\abs{\alpha-x}_p\le p^{-i}[/$]
Page: 12
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(x\in\mathbb Q\)
\(i\in\mathbb N\)
Kind: Lem
Prefix Greater: there exists an integer \(\alpha\) such that
Greater1: \(0\le\alpha<p^i\)
Context: \(p\)-adic
Construction: take \(am\), with \(m\) minimal such that \(mb+np^i=1\)

Definition: The Cauchy sequence \(\{\sum_{i=n}^{m}\{b_i\}\}_m\)
Page: 13
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(n\in\mathbb Z\)
\(\{b_i\}_{i=n}^\infty\in\{0,\dots,p-1\}\)
Notation: \(\sum_{i=n}^\infty\{b_i\}\)

Name: A \(p\)-adic integer
Definition: Some \(a\in\mathbb Q_p\) such that [$]\abs{a}_p\le 1[/$]
Page: 13
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime

Page: 13
Chapter: 1
Section: 4
Name: The ring of \(p\)-adic integer
Set: Cauchy-sequence of integers with [$]\abs{}_p[/$]-norm
Variables: \(p\in\mathbb N\) prime
Notation: \(\mathbb Z_p\)
*: Point-wise product
+: Point-wise sum
0: Constant 0
1: Constant 1
-: Point-wise minus

Name: A rational integer
Definition: An element of \(\mathbb Z\)
Page: 13
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
Context: p-adic number

Page: 13
Definition: [$]\abs{a-b}_p\le p^{-n}[/$]
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
\(a,b\in\mathbb Q_p\)
\(n\in\mathbb Z\)
Notation: \(a\equiv b(\mod p^n)\)
Definition2: \((a-b)/p^n\in\mathbb Z_p\)
Definition3: the first nonzero digit in the \(p\)-adic expansion of \(a - b\) occurs no sooner than the \(p^n\)-place

Name: a \(p\)-adic unit
Definition: \(1/x\in\mathbb Z_p\)
Page: 13
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
Definition2: x\not\equiv0\mod p
Conjdef: The set of \(p\)-adic integers such that, equivalently
Definition3: [$]\abs{x}_p=1[/$]
Definition4: whose first digit in its \(p\)-adic extansion is nonzero

Name: The set of \(p\)-adic units
Page: 13
Chapter: 1
Section: 4
Variables: \(p\in\mathbb N\) prime
Notation: \(\mathbb Z_p^\times\)