Let XK denote the space of
orderings of a field K, and rL∕K: XL→ XK the restriction mapping, when L∕K is a
field extension. Fixing K, the image sets rL∕K(XL) for finite extensions L∕K are
investigated. If K is hilbertian, any clopen subset U ⊂ XK has the form
U = rL∕K(XL) 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.
