Spaces of orderings of fields under finite extensions (Q810088)

Let \(L\supset K\), [L : K]\(<\infty\), be ordered fields with spaces of orders \(X_ L\) and \(X_ K\) and let r: \(X_ L\to X_ K\) be the restriction map. The author studies this situation from the point of view of \textit{M. Marshall's} (abstract) spaces of orderings [Can. J. Math. 31, 320-330 (1979; Zbl 0412.10012)] and [Trans. Am. Math. Soc. 258, 505-521 (1980; Zbl 0427.10015)]. For example he shows i.a.: For any x in \(X_ K\) there is a clopen subspace V of \(X_ K\), with \(x\in V\), and a morphism p of spaces of orderings, p: \(r^{-1}(V)\to r^{-1}(x)\) such that \(p|_{r^{- 1}(x)}\) is the identity and for all \(y\in V\) the restriction p: \(r^{- 1}(y)\to r^{-1}(x)\) is bijective. If in addition L/K is Galois with group G, then p and V can be chosen to be G-invariant. If K is a function field over a real closed field, then the set \(\{x\in X_ k|\) \(r^{- 1}(x)\) is SAP\(\}\) is open and dense in \(X_ K\). Again let L/K be Galois with group G. Let d be the maximal \({\mathbb{Z}}/2{\mathbb{Z}}\)-dimension of an elementary abelian 2-group of G. Then, up to torsion, the trace form of L/K is a multiple of a d-fold Pfister form over K. The methods of proof are valuation-theoretic.