Past exam of the options II course of the University of Oxford / 2012 / A / A2 / c / Solution

Induct through the finitely many construction steps. Initially all available coordinates lie in $\mathbb{Q}$. If coordinates obtained so far lie in a real field $F$, the line through two of its points has coefficients in $F$. A circle with centre in $F^2$ and radius a distance between two such points has equationwhose coefficients lie in $F$, since the squared distance $r^2$ lies in $F$; the radius itself need not lie there.

Part (b) therefore puts every new intersection in $F$ or a real quadratic extension of $F$. If a step produces several points, adjoin their coordinates successively, omitting extensions that add nothing. A finite construction ends in a real field $K$ containing the desired coordinates, with a tower of genuine quadratic extensionsThe tower law for field extensions yieldsA construction involving no nontrivial extension has $n=0$ and degree one.