Example: Examples of Absolute Values

(related to Part: Ordered Fields and Their Topology)

Trivial Absolute Value

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.

P-adic 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.

Absolute Value of Rational Numbers

see absolute value of rational numbers.

Absolute Value of Real Numbers

see absolute value of real numbers.

Absolute Value of Complex Numbers

see absolute value of complex numbers.

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  1. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition