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 $\struct {\Q_p, \norm {\,\cdot\,}_p}$ 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

 * P-adic Integers Form Principal Ideal Domain


 * : $\S 1.8$ Algebraic properties of $p$-adic integers: Proposition $1.46$

Index Laws for Field
User:Leigh.Samphier/P-adicNumbers/Definition:Power of Element

User:Leigh.Samphier/P-adicNumbers/Definition:Power of Element/Field

User:Leigh.Samphier/P-adicNumbers/Definition:Power of Field Element

User:Leigh.Samphier/P-adicNumbers/Powers of Field Elements Commute

User:Leigh.Samphier/P-adicNumbers/Index Laws/Negative Index/Field

User:Leigh.Samphier/P-adicNumbers/Index Laws/Common Index/Field

User:Leigh.Samphier/P-adicNumbers/Index Laws/Sum of Indices/Field

User:Leigh.Samphier/P-adicNumbers/Index Laws/Product of Indices/Field

User:Leigh.Samphier/P-adicNumbers/Index Laws for Field

User:Leigh.Samphier/P-adicNumbers/Common Index Law for Field

User:Leigh.Samphier/P-adicNumbers/Negative Index Law for Field

User:Leigh.Samphier/P-adicNumbers/Sum of Indices Law for Field

User:Leigh.Samphier/P-adicNumbers/Product of Indices Law for Field

Roots of Unity
User:Leigh.Samphier/P-adicNumbers/Definition:Root of Unity

User:Leigh.Samphier/P-adicNumbers/Definition:Root of Unity/Primitive

User:Leigh.Samphier/P-adicNumbers/Definition:Root of Unity/Primitive/Definition 1

User:Leigh.Samphier/P-adicNumbers/Definition:Root of Unity/Primitive/Definition 2

User:Leigh.Samphier/P-adicNumbers/Equivalence of Definitions of Primitive Root of Unity

User:Leigh.Samphier/P-adicNumbers/Root of Unity is Primitive Root for Smaller Power

User:Leigh.Samphier/P-adicNumbers/Integer Power of Root of Unity is Root of Unity

User:Leigh.Samphier/P-adicNumbers/Congruent Powers of Root of Unity are Equal

User:Leigh.Samphier/P-adicNumbers/Power of Root of Unity Equals Power of Remainder

User:Leigh.Samphier/P-adicNumbers/Power of Primitive Root of Unity is Primitive Root of Unity for Divisor

User:Leigh.Samphier/P-adicNumbers/Group of All n-th Roots of Unity

User:Leigh.Samphier/P-adicNumbers/Group of All Roots of Unity

P-adic Numbers
User:Leigh.Samphier/P-adicNumbers/Cyclic Subgroup of P-adic Units formed from (p-1)-th Roots of Unity

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

User:Leigh.Samphier/P-adicNumbers/Cyclic Subgroup of P-adic Units formed from (p-1)-th Roots of Unity

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

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

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


 * Ostrowski's Theorem


 * Product Formula for Norms on Non-zero Rationals

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

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


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

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

User:Leigh.Samphier/P-adicNumbers/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/P-adicNumbers/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/P-adicNumbers/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/P-adicNumbers/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 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/P-adicNumbers/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/P-adicNumbers/2-adic Integers are Homeomorphic to Cantor Set

User:Leigh.Samphier/P-adicNumbers/Cantor Set is Totally Disconnected


 * Cantor Space is Totally Separated


 * Definition:Everywhere Dense


 * Definition:Nowhere Dense


 * Cantor Space is Nowhere Dense

User:Leigh.Samphier/P-adicNumbers/Cantor-like Set

User:Leigh.Samphier/P-adicNumbers/Definition:Cantor-like Set

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

User:Leigh.Samphier/P-adicNumbers/Compact Perfect Totally Disconnected Subset of Real Line is Homeomorphic to Cantor-like Set

User:Leigh.Samphier/P-adicNumbers/P-adic Numbers is Homeomorphic to 2-adic Numbers

User:Leigh.Samphier/P-adicNumbers/Continuous Mapping of Unit Interval onto Unit Square

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers/Examples

User:Leigh/Samphier/P-adicNumbers/Category:Definitions/Examples of Euclidean Models of P-adic Integers

User:Leigh/Samphier/P-adicNumbers/Category:Examples of Euclidean Models of P-adic Integers

User:Leigh/Samphier/P-adic Numbers/Definition:Euclidean Models of P-adic Integers/Examples

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers/Examples/One-dimensional Model of P-adic Integers

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers/Examples/Cantor-like Set is One-dimensional Model of P-adic Integers

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers/Examples/N-dimensional Model of P-adic Integers

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers/Examples/Two-dimensional Model of 3-adic Integers

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers/Examples/Sierpinski gasket

User:Leigh/Samphier/P-adicNumbers/Definition:Euclidean Models of P-adic Integers/Examples/Two-dimensional Model of P-adic Integers

Continuing Fernando Q. Gouvea Book

 * Hensel's Lemma for P-adic Integers


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


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


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

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