Structure of the basic part of a field
For \(a\in K^{\times}\) let \(V_ K(a)\) denote the value group of the form (1,a). Put \(X_ 1=V_ K(a)\), \(X_ 2=\cup \{V_ K(-x):\quad x\in X_ 1\setminus K^{\times 2}\}.\) An element \(a\in K^{\times}\), \(a\not\in \pm K^{\times 2}\) is called basic if (1,a) represents more than two square classes, the set \[ {\mathfrak B}_ K=\pm K^{\times 2}\cup \{a\in K^{\times}:\quad a\quad or\quad -a\quad is\quad basic\} \] is called the ''basic part'' of K. Every \(a\in K^{\times}\setminus {\mathfrak B}_ K\) is ''birigid''. After giving a simple description of the set \(X_ 2=X_ 2(a)\) the author derives the following results: (1) For any global field K (of characteristic not 2) and any \(a\in K^{\times}\) we have \(K^{\times}=X_ 2(a)\). (2) For any purely transcendental extension K of a field F and any \(a\in K^{\times}\) we have \(K^{\times}=X_ 2(a)={\mathfrak B}_ K\). (3) For any \(K\subset {\mathbb{R}}\) with \(| K^{\times}/K^{\times 2}| \geq 4\) and any \(a<0,\) \(-a\not\in K^{\times 2}\) we have \(K^{\times}=X_ 2\cup -X_ 2.\) If \(| K^{\times}/K^{\times 2}| \geq 8\) then \(K^{\times}={\mathfrak B}_ K\). A slightly weaker result holds for \(a>0\).
scientific article; zbMATH DE number 3941619