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

 * Characterisation of Terminal P-adic Expansion


 * : $\S 1.6$ The $p$-adic expansion of rational numbers: Theorem $1.38$

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


 * Hensel's Lemma for P-adic Integers