Most field theory textbooks will describe the field of constructible numbers, i.e. complex numbers corresponding to points in the Euclidean plane that can be constructed via straightedge and compass. This field is the smallest field of characteristic 0 that is closed under square root (i.e. is Pythagorean) and is closed under conjugation.

I'm interested in know: What is the field of numbers that can be constructed if we disallow compass and use only straightedge?

I have not checked this up, but it seems that this question led Hilbert to formulate his 17th problem, particularly the version involving polynomials with rational coefficients (rather than the real coefficients which Artin proved). I'm also interested in knowing more about this history too.

canconstruct in your setting. – user9072 Apr 8 '11 at 13:23markedstraightedge and a compas, then you can trisect and angle: en.wikipedia.org/wiki/Angle_trisection#With_a_marked_ruler – Theo Johnson-Freyd Apr 8 '11 at 14:34