User:Leigh.Samphier/Todo

P-adic last links

 * : $\S 3.1$ Absolute Values on $\Q$
 * Characterisation of Cauchy Sequence in Non-Archimedean Norm


 * : $\S 1.4$ The field of $p$-adic numbers $\Q_p$
 * Definition:P-adic Number

Common
Bold

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 $p$.

Next
Tidy Normed Division Ring Operations are Continuous

Create necessary/sufficient pages for Equivalent Norms on Rational Numbers

Rename pages Valuation Ring of Non-Archimedean Division Ring is Subring

Rename pages Valuation Ideal is Maximal Ideal of Induced Valuation Ring

Create second proof to P-adic Norm not Complete on Rational Numbers

 * Using Hensel's Lemma/First Form

Create third proof to P-adic Norm not Complete on Rational Numbers

 * $\Q_p$ is uncountable, $\Q$ is countable.

Add Definition:Normed Division Algebra to Norm

 * Definition:Normed Division Algebra

Replaced Pages
Leigh.Samphier/Sandbox/Norm Refactor/Normed Division Algebra replaces Definition:Normed Division Algebra

New Pages
Leigh.Samphier/Sandbox/Norm Refactor/Norm/Division Algebra becomes Definition:Norm/Division Algebra

Further Ideas
Definition:Stone Space Stone's Representation Theorem for Boolean Algebras

Definition:Frames & Locales

Nagata-Smirnov Metrization Theorem

Stone-Weierstrass Theorem

Gelfand-Naimark Theorem

Jordan Curve Theorem

Gelfand-Mazur Theorem