Valuation Ideal of P-adic Numbers

Theorem

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Then the valuation ideal induced by norm $\norm {\,\cdot\,}_p$ is the principal ideal:

$p \Z_p = \set {x \in \Q_p: \norm x_p < 1}$

where $\Z_p$ denotes the $p$-adic integers.

Proof

From P-adic Integers is Local Ring, $\Z_p$ is a local ring.

From Principal Ideal from Element in Center of Ring, $p \Z_p$ is a principal ideal.

Now:

\(\ds \norm x_p\)

\(<\)

\(\ds 1\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \norm x_p\)

\(\le\)

\(\ds \dfrac 1 p\)

P-adic Norm of p-adic Number is Power of p

\(\ds \leadstoandfrom \ \ \)

\(\ds p \norm x_p\)

\(\le\)

\(\ds 1\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \dfrac {\norm x_p} {\norm p_p}\)

\(\le\)

\(\ds 1\)

as ${\norm p_p} = \dfrac 1 p$

\(\ds \leadstoandfrom \ \ \)

\(\ds \norm {\dfrac x p}_p\)

\(\le\)

\(\ds 1\)

Norm of Quotient

\(\ds \leadstoandfrom \ \ \)

\(\ds \dfrac x p\)

\(\in\)

\(\ds \Z_p\)

Definition of $p$-adic Integer

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds p \Z_p\)

Hence:

$p \Z_p = \set {x \in \Q_p: \norm x_p < 1}$

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$: Lemma $3.3.4$