(related to Part: Ordered Fields and Their Topology)
Let $F$ be a field. The constant function $|\cdot|:F\to\mathbb R$ with $|x|=1$ for all $x\neq 0$ is an absolute value and a valuation on $F$. It is called the trivial absolute value.
Let $m,n$ be integers $\neq 0$ and let $r,s$ be the highest powers of a prime number $p$ dividing $p^r\mid m$ and $p^s\mid n$ respectively. For the rational number $\frac mn$ we define the p-adic absolute value $$\left|\frac mn\right|_p=p^{r-s},$$ with an integer $r\in\mathbb Z.$ Thus, $|0|_p=0$ and the other axioms of absolute value are easily verified.
see absolute value of rational numbers.
see absolute value of real numbers.
see absolute value of complex numbers.
