# Leigh.Samphier/Sandbox

RED

## Contents

## Continuing Svetlana Book

**Coherent Sequence is Partial Sum of P-adic Expansion

Leigh.Samphier/Sandbox/Partial Sums of P-adic Expansion forms Coherent Sequence

**Leigh.Samphier/Sandbox/Definition:P-adic Numbers as Quotient of Cauchy Sequences

Leigh.Samphier/Sandbox/Integers Arbitrarily Close to Rationals in Valuation Ring of P-adic Norm

Leigh.Samphier/Sandbox/Equivalence Class in P-adic Integers Contains Unique Coherent Sequence

Leigh.Samphier/Sandbox/Equivalence Class in P-adic Integers Contains Unique P-adic Expansion

Leigh.Samphier/Sandbox/Equivalence Class in P-adic Numbers Contains Unique P-adic Expansion

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

Leigh.Samphier/Sandbox/P-adic Norm is Index of First Nonzero Coefficient in P-adic Expansion

Leigh.Samphier/Sandbox/Definition: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

## Matroids

### Loops and Parallel Elements

To be completed.

### Properties of Independent Sets and Bases

Leigh.Samphier/Sandbox/Independent Set can be Augmented by Larger Independent Set

Leigh.Samphier/Sandbox/All Bases of Matroid have same Cardinality

Leigh.Samphier/Sandbox/Matroid satisfies Base Axiom

Leigh.Samphier/Sandbox/Alternative Axiomatization of Matroid