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

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

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

Leigh.Samphier/Sandbox/Power Function on Base Greater than One is Unbounded Above

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

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

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition/Proof 1

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Necessary Condition/Proof 2

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

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

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

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

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

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

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

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

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

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

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

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 11

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 12

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 13

Leigh.Samphier/Sandbox/Canonical P-adic Expansion of Rational is Eventually Periodic/Lemma 14


 * Characterization of Rational P-adic Integer


 * Characterization of Rational P-adic Unit

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