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:

\(\displaystyle x\) \(\in\) \(\displaystyle \map {B^{\,-}_{p^{-n} } } a\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \norm {x - a}_p\) \(\le\) \(\displaystyle p^{-n}\) Definition of Closed Ball in P-adic Numbers
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle p^n \norm {x - a}_p\) \(\le\) \(\displaystyle 1\) Multiply both sides of equation by $p^n$
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \norm {p^{-n} }_p \norm {x - a}_p\) \(\le\) \(\displaystyle 1\) Definition of $p$-adic norm
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \norm {p^{-n} \paren {x - a} }_p\) \(\le\) \(\displaystyle 1\) Norm Axiom $\text N 2$: Multiplicativity
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle p^{-n} \paren {x - a}\) \(\in\) \(\displaystyle \Z_p\) Definition of $p$-adic integers
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x - a\) \(\in\) \(\displaystyle p^n \Z_p\) Definition of Principal Ideal
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle a + p^n \Z_p\) Definition of Left Coset


From set equality:

$\map {B^-_{p^{-n} } } a = a + p^n \Z_p$

The result follows.

$\blacksquare$


Also see