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.

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

Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $\Z^\times_p$ be the $p$-adic units for some prime $p$.

Continuing Svetlana Katok Book

 * Hensel's Lemma for P-adic Integers


 * : $\S 1.7$ Hensel's Lemma and Congruences: Theorem $1.39$

User:Leigh.Samphier/Characterization of Integer Polynomial has Root in P-adic Integers

User:Leigh.Samphier/Characterization of Integer has Square Root in P-adic Integers

User:Leigh.Samphier/Characterization of P-adic Unit has Square Root in P-adic Units exercise 36

User:Leigh.Samphier/Quotient Group of Quaratic Residues Modulo p of P-adic Units exercise 37 and exercise 44


 * Definition:Ideal of Ring


 * Integral Ideal is Ideal of Ring


 * Definition:Maximal Ideal of Ring


 * Prime Element iff Generates Principal Prime Ideal


 * Definition:Quotient Ring


 * Maximal Ideal iff Quotient Ring is Field


 * Definition:Set of Residue Classes


 * Field has Prime Characteristic p iff exists Monomorphism from Field of Integers Modulo p


 * Definition:Integral Domain


 * P-adic Integers Form Integral Domain

User:Leigh.Samphier/Valuation Ideal of P-adic Integers is Unique Maximal Ideal


 * P-adic Integers Form Principal Ideal Domain

User:Leigh.Samphier/Characterization of Primitive m-th Root of Unity Exists in P-adic Numbers

User:Leigh.Samphier/Definition:Signum Function on P-adic Integers

User:leigh.Samphier/Signum Function of P-adic Integers is Well-defined

User:leigh.Samphier/Properties of Signum Function on P-adic Integers


 * Ostrowski's Theorem


 * Product Formula for Norms on Non-zero Rationals

User:Leigh.Samphier/Characterization of Rational Number has Square Root


 * Definition:Open Ball in P-adic Numbers


 * Metric Induces Topology


 * Metric Induces Topology


 * Definition:Topological Subspace


 * Metric Subspace Induces Subspace Topology


 * Definition:Sphere in P-adic Numbers


 * Sphere is Disjoint Union of Open Balls in P-adic Numbers

User:Leigh.Samphier/Sphere is Disjoint Union of Open Balls in P-adic Numbers/Corollary

User:Leigh.Samphier/Sphere is Open in P-adic Numbers


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

User:Leigh.Samphier/Sphere is Not Boundary of Open Ball in P-adic Numbers

User:Leigh.Samphier/Closed Ball is Not Closure of Open Ball in P-adic Numbers


 * Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls

User:Leigh.Samphier/Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls/Corollary


 * Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls

User:Leigh.Samphier/Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls/Corollary


 * Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen

User:Leigh.Samphier/Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen/Corollary


 * Countable Basis for P-adic Numbers


 * Sphere is Disjoint Union of Open Balls in P-adic Numbers


 * Definition:Sequentially Compact Space


 * Definition:Compact Metric Space


 * P-adic Integers are Compact Subspace


 * P-adic Integers are Compact Subspace


 * P-adic Numbers is Locally Compact Topological Space


 * P-adic Integers is Metric Completion of Integers


 * Definition:Disconnected (Topology)


 * Definition:Connected (Topology)


 * Definition:Totally Disconnected Space


 * P-adic Numbers is Totally Disconnected Topological Space

User:Leigh.Samphier/Multiplicative Subgroup of Quaratic Residues Modulo p of P-adic Units is Open


 * Definition:Cantor Set/Limit of Decreasing Sequence Create Definition:Cantor Set as Limit of Decreasing Sequence


 * Equivalence of Definitions of Cantor Set


 * Cantor Set is Uncountable


 * Cantor Space is Perfect


 * Definition:Continuous Mapping (Topology)


 * Definition:Open Mapping


 * Definition:Homeomorphism/Metric Spaces/Definition 1


 * Definition:Continuous Mapping (Topology)/Point/Neighborhoods


 * Definition:Uniform Continuity/Metric Space


 * Definition:Isometry (Metric Spaces)

User:Leigh.Samphier/Dyadic Integers are Homeomorphic to Cantor Set

User:Leigh.Samphier/Cantor Set is Totally Disconnected


 * Cantor Space is Totally Separated


 * Definition:Everywhere Dense


 * Definition:Nowhere Dense


 * Cantor Space is Nowhere Dense

User:Leigh.Samphier/Cantor-like Set

User:Leigh.Samphier/P-adic Numbers are Homeomorphic to Cantor-like Set

Continuing Fernando Q. Gouvea Book

 * Hensel's Lemma for P-adic Integers


 * : $\S 3.4$ Hensel's Lemma $\Q_p$, Theorem $3.4.1$


 * User:Leigh.Samphier/Characterization of Primitive m-th Root of Unity Exists in P-adic Numbers


 * User:Leigh.Samphier/Characterization of P-adic Unit has Square Root in P-adic Units

User:Leigh.Samphier/Characterization of P-adic Number has Square Root