- Definitions related to P-adic Number Theory can be found here.
- Results about P-adic Number Theory can be found here.
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
- Definition:Root of Unity/Primitive/Definition 2
Roots of Unity
User:Leigh.Samphier/P-adicNumbers/Root of Unity is Primitive Root for Smaller Power
User:Leigh.Samphier/P-adicNumbers/Power of Primitive Root of Unity is Primitive Root of Unity for Divisor
User:Leigh.Samphier/P-adicNumbers/Cyclic 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
- Definition:Root of Unity/Primitive/Definition 2
- 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