Closed Ball of P-adic Number
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Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ denote the $p$-adic integers.
Let $a \in \Q_p$.
For all $\epsilon \in \R_{>0}$, let $\map { {B_\epsilon}^-} a$ denote the closed ball of $a$ of radius $\epsilon$.
Then:
- $\forall n \in Z : \map {B^-_{p^{-n} } } a = a + p^n \Z_p$
where $a + p^n \Z_p$ denotes the left coset of the principal ideal $p^n \Z_p$ containing $a$ in the subring $\Z_p$.
That is, the closed ball $\map { {B_\epsilon}^-} a$ is the set:
- $a + p^n \Z_p = \set{a + p^n z : z \in \Z_p}$
Proof
Let $n \in \Z$.
Then:
\(\ds x\) | \(\in\) | \(\ds \map {B^{\,-}_{p^{-n} } } a\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \norm {x - a}_p\) | \(\le\) | \(\ds p^{-n}\) | Definition of Closed Ball in P-adic Numbers | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds p^n \norm {x - a}_p\) | \(\le\) | \(\ds 1\) | Multiply both sides of equation by $p^n$ | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \norm {p^{-n} }_p \norm {x - a}_p\) | \(\le\) | \(\ds 1\) | Definition of $p$-adic norm | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \norm {p^{-n} \paren {x - a} }_p\) | \(\le\) | \(\ds 1\) | Non-Archimedean Norm Axiom $\text N 2$: Multiplicativity | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds p^{-n} \paren {x - a}\) | \(\in\) | \(\ds \Z_p\) | Definition of $p$-adic integers | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x - a\) | \(\in\) | \(\ds p^n \Z_p\) | Definition of Principal Ideal | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(\in\) | \(\ds a + p^n \Z_p\) | Definition of Left Coset |
From set equality:
- $\map {B^-_{p^{-n} } } a = a + p^n \Z_p$
The result follows.
$\blacksquare$