# Leigh.Samphier/Sandbox

RED

## Contents

## $p$-adic Numbers

### Continuing Svetlana Katok Book

- 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*... (previous): $\S 1.4$ The field of $p$-adic numbers $\Q_p$

Leigh.Samphier/Sandbox/Complete Normed Division Ring is Completion of Dense Subring

Leigh.Samphier/Sandbox/Distance-Preserving Homomorphism Preserves Norm

Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring

Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring/Corollary

Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion

Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion/Corollary

Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p

Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma

Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Proof 1

Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Proof 2

Leigh.Samphier/Sandbox/P-adic Expansion Representative of P-adic Number is Unique

Leigh.Samphier/Sandbox/Representative of P-adic Sum

Leigh.Samphier/Sandbox/Representative of P-adic Product

Leigh.Samphier/Sandbox/Cauchy Sequence Represents P-adic Number

Leigh.Samphier/Sandbox/Constant Sequence Represents Rational in P-adic Numbers

### Continuing Fernando Q. Gouvea Book

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*: $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$

### Every P-adic Number is Limit of P-adic Expansion

Leigh.Samphier/Sandbox/Sequence Converges in Completion iff Sequence Represents Limit

Leigh.Samphier/Sandbox/Distinct P-adic Expansions Converge to Distinct P-adic Numbers

Leigh.Samphier/Sandbox/P-adic Number is Limit of Unique P-adic Expansion - Complete the uniqueness

### Characterisation of P-adic Units

Leigh.Samphier/Sandbox/Set of P-adic Units is Unit Sphere

Leigh.Samphier/Sandbox/Characterization of Rational P-adic Units

Leigh.Samphier/Sandbox/P-adic Expansion of P-adic Units

### Create theorem P-adic Numbers are Uncountable

## Matroids

### Loops and Parallel Elements

- 1976: Dominic Welsh:
*Matroid Theory*Chapter $1.$ $\S 4.$ Loops and parallel elements

Leigh.Samphier/Sandbox/Doubleton of Elements is Subset

Leigh.Samphier/Sandbox/Power Set of Doubleton

Leigh.Samphier/Sandbox/Distinct Elements are Parallel iff Pair forms Circuit

Leigh.Samphier/Sandbox/Parallel Relationship is Transitive

Leigh.Samphier/Sandbox/Distinct Elements are Parallel iff Each is in Closure of Other

Leigh.Samphier/Sandbox/Closure of Subset Contains Parallel Elements

Leigh.Samphier/Sandbox/Set with Two Parallel Elements is Dependent

Leigh.Samphier/Sandbox/Definition:Simple Matroid

Leigh.Samphier/Sandbox/Loop Belongs to Every Flat

Leigh.Samphier/Sandbox/Parallel Elements Depend on Same Subsets

Leigh.Samphier/Sandbox/Matroid Contains No Loops iff Empty Set is Flat

### Properties of Independent Sets and Bases

- 1976: Dominic Welsh:
*Matroid Theory*Chapter $1.$ $\S 5.$ Properties of independent sets and bases

Leigh.Samphier/Sandbox/Matroid satisfies Base Axiom

Leigh.Samphier/Sandbox/Alternative Axiomatization of Matroid

### Properties of the Rank Function

Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms

### The Closure Operator

### Closed Sets = Flats = Subspaces

### Circuits

### The Cycle Matroid of a Graph

### The Greedy Algorithm

- 1976: Dominic Welsh:
*Matroid Theory*... (previous) Chapter $19.$ $\S 1.$ The greedy algorithm

Maximum Weight Problem (Greedy Algorithm)

Complete Greedy Algorithm yields Maximal Set

Complete Greedy Algorithm may not yield Maximum Weight

Complete Greedy Algorithm guarantees Maximum Weight iff Matroid

## Stone Spaces

- 1982: Peter T. Johnstone:
*Stone spaces*: Chapter $\text I$: Preliminaries, Definition $1.3$