Definition 1.1 (Absolute value). Let $K$ be a field. An absolute value on $K$ is a function $|\bullet |:K\to {ℝ}_{\ge 0}$ such that

• (i) $|x|=0$ if and only if $x=0$.
• (ii) $|xy|=|x||y|$ for all $x,y\in K$.
• (iii) $|x+y|\le |x|+|y| \forall x,y\in K$ (triangly inequality).

We say $(K,|\bullet |)$ is a valued field.