The problem is in defining the $p$-adic norm for $p$ composite.
If $p$ is prime, we can write any nonzero rational number $x$ as $x=\frac{p^a r}{s}$ where $r$ and $s$ are not divisible by $p$. The fact that $p$ is prime guarantees that the exponent $a$ is unique. Then we define the $p$-adic norm of $x$ to be $|x|_p=p^{-a}$.
If $p$ is composite, say $p=4$, then the exponent $a$ is not unique. For instance, take $x=8$. We can write $x=\frac{4^1\cdot 2}{1}= \frac{4^2 \cdot 1}{2}$, and in both instances $r$ and $s$ are not divisible by $4$. But this gives us two possible values of $a$ - either $1$ or $2$. There's just no way to fix this to get a $4$-adic norm.