P-adic Numbers are Uncountable
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Theorem
Let $p$ be any prime number.
The set of $p$-adic numbers $\Q_p$ is an uncountable set.
Proof
Let $P$ be the set of sequences on $\set{i : i \in \N : 0 \le i < p}$.
That is:
- $P = \set{\sequence{d_n} : d_n \in \N : 0 \le d_n < p}$
From Cantor's Diagonal Argument:
- $P$ is an uncountable set
Let $f: P \to \Q_p$ be the mapping from $P$ to $\Z_p$ defined by:
- $\forall \sequence{d_n} \in P : \map f {\sequence{d_n}} = \ds \sum_{n = 0}^\infty d_n p^n$
where $Z_p$ denotes the $p$-adic integers and $\ds \sum_{n = 0}^\infty d_n p^n$ denotes a $p$-adic expansion
From P-adic Integer has Unique P-adic Expansion Representative:
- $f$ is bijective
Hence:
- $\Z_p$ is an uncountable set
Recall that $\Z_p \subseteq \Q_p$.
From Sufficient Conditions for Uncountability:
- $\Q_p$ is an uncountable set
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Exercise $20$