Residue Field of P-adic Norm on Rationals/Lemma 2

Theorem

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $\Z_{\ideal p}$ be the induced valuation ring on $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $p \Z_{\ideal p}$ be the induced valuation ideal on $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\phi : \Z \to \Z_{\ideal p} / p \Z_{\ideal p}$ be the mapping defined by:

$\forall a \in \Z: \map \phi a = a + p \Z_{\ideal p}$

Then:

$p \Z = \map \ker \phi$

Proof

Let $\map \ker \phi$ denote the kernel of $\phi$.

Then:

\(\ds a\)

\(\in\)

\(\ds \map \ker \phi\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \map \phi a\)

\(=\)

\(\ds p \Z_{\ideal p}\)

Definition of Kernel of Ring Homomorphism

\(\ds \leadstoandfrom \ \ \)

\(\ds a + p \Z_{\ideal p}\)

\(=\)

\(\ds p \Z_{\ideal p}\)

Definition of $\phi$

\(\ds \leadstoandfrom \ \ \)

\(\ds a = a - 0\)

\(\in\)

\(\ds p \Z_{\ideal p}\)

Element in Right Coset iff Product with Inverse in Subgroup

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists a' \in \Z: \, \)

\(\ds a = p a'\)

\(=\)

\(\ds p a'\)

\(\ds \leadstoandfrom \ \ \)

\(\ds a\)

\(\in\)

\(\ds p \Z\)

Hence:

$p \Z = \map \ker \phi$

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 2.4$ Algebra, Proposition $2.4.3$