P-adic Norm and Absolute Value are Not Equivalent
Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.
Let $\size{\,\cdot\,}$ be the absolute value on the rationals $\Q$.
Then $\norm {\,\cdot\,}_p$ and $\size{\,\cdot\,}$ are not equivalent norms.
That is, the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topology induced by $\size {\,\cdot\,}$.
Proof 1
By definition of the $p$-adic norm:
- $\norm p_p = \frac 1 p < 1$
By definition of the absolute value:
- $\size p = p > 1$
By definition of open unit ball equivalence, $\norm {\,\cdot\,}_p$ and $\size {\,\cdot\,}$ are not equivalent norms.
By Equivalence of Definitions of Equivalent Division Ring Norms and the definition of topologically equivalent norms then the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topology induced by $\size{\,\cdot\,}$.
$\blacksquare$
Proof 2
It is noted that:
- $\sup \set {\size n: n \in \Z} = +\infty$
By a corollary of Characterisation of Non-Archimedean Division Ring Norms then $\size {\,\cdot\,}$ is Archimedean.
From P-adic Norm on Rational Numbers is Non-Archimedean Norm then $\norm {\,\cdot\,}_p$ is non-Archimedean.
By Equivalent Norms are both Non-Archimedean or both Archimedean, $\norm {\,\cdot\,}_p$ and $\size {\,\cdot\,}$ are not equivalent norms.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.1.2$ Absolute Values on $\Q$, Problem $68$