User:Leigh.Samphier/P-adicNumbers

$p$-adic Numbers
Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring with zero $0_R$ and unity $1_R$.

Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$.

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

Let $\tau$ be the topology induced by the norm $\norm{\,\cdot\,}$.

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Cleanup/Refactor
Leigh.Samphier/Sandbox/Completion of Normed Division Ring

Leigh.Samphier/Sandbox/Definition:P-adic Number


 * P-adic Expansion is a Cauchy Sequence in P-adic Norm/Represents a P-adic Number

Leigh.Samphier/Sandbox/Definition:P-adic Number/Representative

Leigh.Samphier/Sandbox/Representative of P-adic Number is Representative of Equivalence Class

Leigh.Samphier/Sandbox/P-adic Numbers form Non-Archimedean Valued Field


 * Replace P-adic Norm satisfies Non-Archimedean Norm Axioms with Leigh.Samphier/Sandbox/P-adic Numbers form Non-Archimedean Valued Field


 * Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers


 * Characterization of Closed Ball in P-adic Numbers


 * Characterization of Open Ball in P-adic Numbers


 * P-adic Norm satisfies Non-Archimedean Norm Axioms


 * Open and Closed Balls in P-adic Numbers are Clopen in P-adic Metric


 * P-adic Open Ball is Instance of Open Ball of a Norm


 * P-adic Closed Ball is Instance of Closed Ball of a Norm


 * P-adic Sphere is Instance of Sphere of a Norm


 * Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm


 * Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers


 * Local Basis of P-adic Number


 * Equivalence of Definitions of Convergent P-adic Sequence


 * P-adic Norm satisfies Non-Archimedean Norm Axioms


 * P-adic Numbers is Totally Disconnected Topological Space

Leigh.Samphier/Sandbox/P-adic Numbers form Completion of Rational Numbers with P-adic Norm

????? Leigh.Samphier/Sandbox/Completion of Rational Numbers with P-adic Norm


 * P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number

Leigh.Samphier/Sandbox/Rational Numbers form Dense Subfield of P-adic Numbers


 * Integers are Arbitrarily Close to P-adic Integers


 * P-adic Norm of p-adic Number is Power of p


 * Integers are Arbitrarily Close to P-adic Integers


 * Countable Basis for P-adic Numbers

Leigh.Samphier/Sandbox/P-adic Norm on P-adic Numbers Extends Norm on Rational Numbers


 * Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers


 * P-adic Valuation Extends to P-adic Numbers


 * P-adic Metric on P-adic Numbers is Non-Archimedean Metric


 * Valuation Ring of P-adic Norm is Subring of P-adic Integers


 * P-adic Norm of p-adic Number is Power of p

Other

 * Open and Closed Balls in P-adic Numbers are Compact Subspaces


 * Open and Closed Balls in P-adic Numbers are Totally Bounded

Leigh.Samphier/Sandbox/Definition:P-adic Norm

Leigh.Samphier/Sandbox/Definition:P-adic Norm/Rational Numbers

Leigh.Samphier/Sandbox/Definition:P-adic Norm on Rational Numbers

Leigh.Samphier/Sandbox/Definition:P-adic Norm/Rational Numbers/Definition 1

Leigh.Samphier/Sandbox/Definition:P-adic Norm/Rational Numbers/Definition 2

Leigh.Samphier/Sandbox/Equivalence of Definitions of P-adic Norms

Leigh.Samphier/Sandbox/Equivalence of Definitions of P-adic Norms/Lemma 1

Leigh.Samphier/Sandbox/P-adic Norm Characterisation of Divisibility by Power of p

Leigh.Samphier/Sandbox/Definition:P-adic Norm on P-adic Numbers

Leigh.Samphier/Sandbox/Definition:P-adic Norm/P-adic Numbers

Leigh.Samphier/Sandbox/Definition:P-adic Norm/P-adic Numbers/Notation

Leigh.Samphier/Sandbox/Definition:P-adic Metric

Leigh.Samphier/Sandbox/Definition:P-adic Metric/Rational Numbers

Leigh.Samphier/Sandbox/Definition:P-adic Metric on Rational Numbers

Leigh.Samphier/Sandbox/Definition:P-adic Metric/P-adic Numbers

Leigh.Samphier/Sandbox/Metric on P-adic Numbers Extends Metric on Rationals

Leigh.Samphier/Sandbox/Integers with Metric Induced by P-adic Valuation

Leigh.Samphier/Sandbox/Restricted P-adic Metric is Metric

Delete Definition:P-adic Metric/Restricted

Delete Definition:Restricted P-adic Metric/Definition 1

Delete Definition:Restricted P-adic Metric/Definition 2

Continuing Svetlana Katok Book

 * : $\S 1.4$ The field of $p$-adic numbers $\Q_p$
 * P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient

Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism

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

 * : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$
 * P-adic Number is Limit of Unique P-adic Expansion

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