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

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

Let $\phi : \Z \to \Z_{(p)}/p\Z_{(p)}$ be the mapping defined by:
 * $\forall a \in \Z: \map \phi a = \eqclass a {}$

Then:
 * $p\Z = \ker \paren \phi$

Proof
Let $K = \ker \paren{\phi}$ be the kernel of $\phi$.

Then:

Hence $p\Z = K = \ker \paren{\phi}$.