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.

Leigh.Samphier/Sandbox/Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit

Leigh.Samphier/Sandbox/Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Necessary Condition

Leigh.Samphier/Sandbox/Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Sufficient Condition

Leigh.Samphier/Sandbox/Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 1

Leigh.Samphier/Sandbox/Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit/Lemma 2

Leigh.Samphier/Sandbox/Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit

Leigh.Samphier/Sandbox/Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit/P-adic Expansion

Leigh.Samphier/Sandbox/P-adic Expansion Converges to P-adic Number iff P-adic Expansion Represents P-adic Number

Fix Valuation Ideal is Maximal Ideal of Induced Valuation Ring

Fix Definition:Non-Archimedean/Norm (Division Ring)/Archimedean

Fix Definition:Closure Axioms (Matroid)

Continuing Svetlana Katok Book

 * : $\S 1.4$ The field of $p$-adic numbers $\Q_p$
 * P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient

Leigh.Samphier/Sandbox/Definition:P-adic Digits

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

Continuing Fernando Q. Gouvea Book

 * : $\S 3.3$ Exploring $\Q_p$, Lemma $3.3.8$
 * P-adic Number is Limit of Unique P-adic Expansion

Every P-adic Number is Limit of Unique P-adic Expansion
Leigh.Samphier/Sandbox/P-adic Number is Limit of Unique P-adic Expansion

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

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

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