Residue Field of P-adic Norm on Rationals

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

The induced residue field on $\struct {\Q,\norm {\,\cdot\,}_p}$ is isomorphic to the field $\F_p$ of integers modulo $p$.

Proof
By Valuation Ring of P-adic Norm on Rationals,
 * $\Z_{(p)} = \set{ \dfrac a b \in \Q : p \nmid b }$

is the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$.

By Valuation Ideal of P-adic Norm on Rationals,
 * $p\Z_{(p)} = \set{ \dfrac a b \in \Q : p \nmid b, p \divides a}$

is the induced valuation ideal on $\struct {\Q,\norm {\,\cdot\,}_p}$.

By definition, the induced residue field on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the quotient ring $\Z_{(p)}/p\Z_{(p)}$.

By Quotient Ring of Integers with Principal Ideal, $\F_p$ is isomorphic to $\Z/p\Z$, where $p\Z$ is the principal ideal of $\Z$ generated by $p$.

To complete the proof it is sufficient to show that $\Z/p\Z$ is isomorphic to $\Z_{(p)}/p\Z_{(p)}$.

Since $p \nmid 1$ then for all $a \in \Z$, $a = \dfrac a 1 \in \Z_{(p)}$.

Hence $\Z \subset \Z_{(p)}$ is a subring of $\Z_{(p)}$.

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

Lemma 3
Hence $\phi$ is a ring epimorphism with:
 * $p\Z = \ker \paren{\phi}$

By Quotient Ring of Kernel of Ring Epimorphism then $\Z/p\Z$ is isomorphic to $\Z_{(p)}/p\Z_{(p)}$

The result follows.