P-adic Expansion Representative of P-adic Number is Unique

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 $\Q_p$.

Let $\ds \sum_{i \mathop = m}^\infty d_i p^i$ and $\ds \sum_{i \mathop = k}^\infty e_i p^i$ be $p$-adic expansions that represent $a$.

Then:

$(1) \quad m = k$

$(2) \quad \forall i \ge m : d_i = e_i$

That is, the $p$-adic expansions $\ds \sum_{i \mathop = m}^\infty d_i p^i$ and $\ds \sum_{i \mathop = k}^\infty e_i p^i$ are identical.

Proof

From P-adic Number times P-adic Norm is P-adic Unit there exists $n \in \Z$:

$p^n a \in \Z^\times_p$

$\norm a_p = p^n$

where $\Z^\times_p$ is the set of $p$-adic units.

Let $l = -n$.

From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:

$l = \min \set {i: i \ge m \land d_i \ne 0}$

and

$l = \min \set {i: i \ge k \land e_i \ne 0}$

Now:

\(\ds m\)

\(<\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds d_m\)

\(\ne\)

\(\ds 0\)

Definition of $p$-adic Expansion

\(\ds \leadsto \ \ \)

\(\ds m\)

\(=\)

\(\ds l\)

Definition of $l$

\(\ds \leadsto \ \ \)

\(\ds l\)

\(<\)

\(\ds 0\)

Similarly:

$k < 0 \leadsto k = l$

Proof of statement $(1)$

Case $l < 0$

Let $l < 0$.

Then:

\(\ds l\)

\(<\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds m\)

\(<\)

\(\ds 0\)

as $m \le l$

\(\ds \leadsto \ \ \)

\(\ds m\)

\(=\)

\(\ds l\)

from above

Similarly:

$l < 0 \leadsto k = l$

So:

$m = l = k$.

$\Box$

Case $l \ge 0$

Let $l \ge 0$.

Then:

\(\ds l\)

\(\ge\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds m\)

\(\not <\)

\(\ds 0\)

as $m < 0 \leadsto l < 0$

\(\ds \leadsto \ \ \)

\(\ds m\)

\(=\)

\(\ds 0\)

Definition of $p$-adic Expansion

Similarly:

$l \ge 0 \leadsto k = 0$

So:

$m = 0 = k$.

$\Box$

Proof of statement $(2)$

By definition of $l$, for all $i$ such that $m \le i < l$:

$d_i = 0$

$e_i = 0$

So for all $m \le i < l$:

$d_i = e_i$.

Consider the series:

$\ds \sum_{i \mathop = l}^\infty d_i p^i$

From P-adic Expansion Less Intial Zero Terms Represents Same P-adic Number

$\ds \sum_{i \mathop = l}^\infty d_i p^i$

is a representative of $a$

From Multiple Rule for Cauchy Sequences in Normed Division Ring:

$\ds p^{-l} \sum_{i \mathop = l}^\infty d_i p^i = \sum_{i \mathop = l}^\infty d_i p^{i - l} = \sum_{i \mathop = 0}^\infty d_{i + l} p^i$

is a representative of $p^{-l} a \in \Z^\times_p$.

By definition of a $p$-adic expansion:

$\forall i \in N : 0 \le d_{i + 1} < p - 1$

Thus the series:

$\ds \sum_{i \mathop = 0}^\infty d_{i + l} p^i$

is a $p$-adic expansion that represents $p^{-l} a \in \Z^\times_p$.

Similarly the series:

$\ds \sum_{i \mathop = 0}^\infty e_{i + l} p^i$

is a $p$-adic expansion that represents $p^{-l} a \in \Z^\times_p$.

From P-adic Integer has Unique P-adic Expansion Representative it follows that:

$\forall i \in N : d_{i + 1} = e_{i + 1}$

That is:

$\forall i \ge l : d_i = e_i$

The result follows.

$\blacksquare$