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

 * : $\S 1.4$ The field of $p$-adic numbers $\Q_p$
 * P-adic Number times P-adic Norm is P-adic Unit

User:Leigh.Samphier/Sandbox/Definition:Periodic P-adic Expansion

User:Leigh.Samphier/Sandbox/Definition:Eventually Periodic P-adic Expansion

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Sufficient Condition

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 1

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 2

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 3

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 4

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 5

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 6

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 7

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 8

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 9

User:Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 10


 * Characterization of Rational P-adic Integer


 * Characterization of Rational P-adic Unit


 * User:Leigh.Samphier/Sandbox/Definition:Terminal P-adic Expansion


 * User:Leigh.Samphier/Sandbox/Characterisation of Terminal P-adic Expansion

User:Leigh.Samphier/Sandbox/Characterisation of Terminal P-adic Expansion/Necessary Condition

User:Leigh.Samphier/Sandbox/Characterisation of Terminal P-adic Expansion/Sufficient Condition


 * P-adic Integers Form Integral Domain


 * P-adic Number is Power of p Times P-adic Unit


 * Ideals of P-adic Integers


 * Ideals of P-adic Integers/Corollary


 * P-adic Integers Form Principal Ideal Domain


 * Closed Subgroups of P-adic Integers


 * Polynomials of Congruent Ring Elements are Congruent


 * Congruence Modulo a Principal Ideal of P-adic Integers

User:Leigh.Samphier/Sandbox/Congruence Modulo Equivalence for Integers in P-adic Integers

User:Leigh.Samphier/Sandbox/Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1

User:Leigh.Samphier/Sandbox/Hensel's Lemma

User:Leigh.Samphier/Sandbox/Hensel's Lemma for P-adic Integers

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 1

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 2

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 3

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 4

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 5

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 6

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 7

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 8

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 9

User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Lemma 10

Continuing Fernando Q. Gouvea Book

 * : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$
 * Sequence of P-adic Integers has Convergent Subsequence/Proof 2


 * User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers