P-adic Number has Unique P-adic Expansion Representative
Theorem
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Then $a$ has exactly one representative that is a $p$-adic expansion.
Proof
Case 1
Let $\norm a_p \le 1$.
From P-adic Integer has Unique P-adic Expansion Representative, $a$ has exactly one representative that is a $p$-adic expansion of the form:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$
$\Box$
Case 2
Let $\norm a_p > 1$.
From P-adic Number times P-adic Norm is P-adic Unit,
- $\exists m \in \Z: p^m a \in \Z_p^\times$
where $\Z_p^\times$ denotes the $p$-adic units.
By definition a $p$-adic unit is a $p$-adic integer.
From P-adic Integer has Unique P-adic Expansion Representative:
- $p^m a$ has exactly one representative that is a $p$-adic expansion of the form:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$
From P-adic Unit has Norm Equal to One:
- $\norm {p^m a}_p = 1 = p^0$
From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:
- $d_0 \neq 0$
Let $\sequence{x_n}$ be a representative of $a \in \Q_p$.
From Rational Numbers are Dense Subfield of P-adic Numbers:
- the constant sequence $\tuple {p^m, p^m, p^m, \dotsc}$ represents $p^m \in \Q_p$.
By definition of the product in a quotient ring:
- the Cauchy sequence $\sequence{p^m x_n}$ is a representative of $p^m a \in \Q_p$.
From Representatives of same P-adic Number iff Difference is Null Sequence:
- $\sequence {p^m x_n - \ds \sum_{i \mathop = 0}^n d_i p^i}$ is a null sequence
From Multiple Rule for Cauchy Sequences in Normed Division Ring:
- $\sequence {x_n - \ds p^{-m} \sum_{i \mathop = 0}^n d_i p^i}$ is a null sequence
From Representatives of same P-adic Number iff Difference is Null Sequence:
- $\sequence {\ds p^{-m} \sum_{i \mathop = 0}^\infty d_i p^i}$ is a representative of $a$.
That is:
- $\ds \sum_{i \mathop = -m}^\infty e_i p^i$ is a representative of $a$
where:
- $\forall i \ge -m: e_i = d_{i + m}$
By definition of a $p$-adic expansion;
- $\forall i \ge 0: 0 \le d_i < p$
Then:
- $\forall i \ge -m: 0 \le e_i = d_{i + m} < p$
Since $d_0 \neq 0$, then $e_{-m} = d_0 \neq 0$.
By definition of a $p$-adic expansion;
- $\ds \sum_{i \mathop = -m}^\infty e_i p^i$ is a $p$-adic expansion that represents $a$
From P-adic Expansion Representative of P-adic Number is Unique:
- $\ds \sum_{i \mathop = -m}^\infty e_i p^i$ is the only $p$-adic expansion that represents $a$
$\blacksquare$
Also see
- P-adic Number is Limit of Unique P-adic Expansion
- P-adic Integer has Unique Coherent Sequence Representative
- P-adic Integer has Unique P-adic Expansion Representative
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$