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 $p$.

FIX
Valuation Ideal is Maximal Ideal of Induced Valuation Ring

Continuing Svetlana Katok Book

 * : $\S 1.4$ The field of $p$-adic numbers $\Q_p$
 * Equivalence of Definitions of P-adic Integer

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Proof 1

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Proof 2

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 1

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 2

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 3

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 4

Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence/Lemma 5

Leigh.Samphier/Sandbox/P-adic Unit has Non-Zero First P-adic Digit


 * Definition:P-adic Unit


 * P-adic Unit has Norm Equal to One


 * P-adic Number times P-adic Norm is P-adic Unit

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic to the Left

Leigh.Samphier/Sandbox/Characterization of Rational P-adic Integer


 * Leigh.Samphier/Sandbox/Characterization of Rational P-adic Unit

Continuing Fernando Q. Gouvea Book

 * : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$
 * P-adic Unit has Norm Equal to One


 * Leigh.Samphier/Sandbox/Characterization of Rational P-adic Unit Lemma 3.3.4 (i) link between Lemma 3.3.4 and Lemma 3.3.4 (ii)

Leigh.Samphier/Sandbox/Set of P-adic Units is Unit Sphere

Leigh.Samphier/Sandbox/Characterization of Rational P-adic Unit

Leigh.Samphier/Sandbox/P-adic Expansion of P-adic Unit


 * Leigh.Samphier/Sandbox/Sequence of P-adic Integers has Convergent Subsequence