Some remarks on orderings under finite field extensions
Claus Scheiderer · Pacific Journal of Mathematics · 1992
Let XK denote the space of orderings of a field K, and r L / K : XL -• XK the restriction mapping, when L/K is a field extension.Fixing K, the image sets r L/κ (X L ) for finite extensions L/K are investigated.If K is hilbertian, any clopen subset U C XK has the form U = r L / K {Xι) for some finite L/K, and [L : K] can be bounded in terms of U .This bound is even sharp in some cases, but not always.A second construction gives the same qualitative result for a much larger class of fields.It is based on iterated quadratic extensions.The bounds on [L : K] obtained here are weaker than in the hilbertian case.