Approximation theorems for topological, valued, and ordered nearfields (Q1338022)

The author investigates to what extent the classical approximation theorem for valuated (commutative) fields can be proved in nearfields. As known, even skewfields possess different notions for valuations. An approximation theorem can only be shown for a relatively general concept of a valuation. The author uses here the following definition: Let \(F\) be a nearfield and \(v: F\to (\Gamma,\leq)\) a mapping of \(F\) into a totally ordered set \(\Gamma\) which has a smallest element \(\Theta\) and satisfies: \[ \begin{aligned} &\text{B}1: \quad v(x)= \Theta \iff x=0, \\ &\text{B}2: \quad v(x+y)\leq \max\{ v(x), v(y)\},\\ &\text{B}3: \quad v(x)\leq v(y) \Rightarrow v(ax)\leq v(ay) \text{ for any }a\in F. \end{aligned} \] A valuation \(v\) is a \(V\)-valuation if in addition the axiom (VB) is valid: \[ \begin{aligned} \text{(VB)} \quad &\text{For any } \gamma\in v(F^*) \subseteq \Gamma \text{ there exists }\gamma'\in v(F^*)\\ &\text{such that }v(x)> \gamma' \text{ implies }v(x^{-1})< \gamma. \end{aligned} \] The approximation theorem for valued nearfields now reads as follows: Let \(v_1, v_2, \dots, v_n\) be pairwise topologically inequivalent \(V\)-valuations on a nearfield \(F\) and let \(x_i\in F\) and \(\gamma_i\in v_i (F^*)\), \(i=1, \dots, n\), be arbitrarily chosen. Then there exists an element \(a\in F\) such that \(v_i (x_i- a)< \gamma_i\) for \(i=1,2, \dots,n\). As an application the author investigates also which orderings on nearfields induce \(V\)-valuations. This is the case for \(V\)-orders, i.e. orders satisfying: \[ \text{For any \(a>0\) there exists \(a'>0\) such that \(a'<x\in F\) implies }x^{-1}<a. \tag{VO} \] The \(V\)-orders possess the advantage that the order-topology coincides with the valuation topology of the induced valuation. For \(4\)-orders an analogous approximation theorem is proved.