For a general normed space $(V,\|{\cdot}\|)$ suppose that the vector space $V$ is over some field $K$ that is not $\Bbb Q$, $\Bbb R$, $\Bbb C$, $\Bbb Z_p$ or any of it expansions, then, in general, how we define the absolute value such that
$$\|\alpha x\|=|\alpha|\|x\|$$
where $\alpha\in K$? I suppose we will need to define some function of the kind
$$|{\cdot}|:K\to\Bbb R_{\ge 0}$$
I searched some information about the general topic of normed spaces but I dont found something about this, can you give me some text, link, bibliography or just enlighten this question? Thank you.
The absolute value is a notion related to vector lattices (which has nothing to do with the notion of norm). Indeed, given a vector space $X$ with a partial order $\le$ such that finite suprema (and infima) exist, then the absolute value is defined by $$ |x|:=\sup\{x,-x\}. $$ for each $x \in X$. For a standard textbook, you can see Positive Operators.