History of the three "impossible" compass-and-straightedge problems I'm preparing a presentation about constructible numbers and I wanted to know some of the history about it to motivate the topic.
I wanted to know if the classical Greek problems (doubling the cube, squaring the circle and the angle trisection) were first proven to be impossible using Field Theory or there is another method that was developed earlier to prove this.
 A: Sites around the internet give Pierre Laurent Wantzel credit for proving these problems as impossible in the paper "Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas" in "Liouville's Journal" (1837).

"In Wantzel's paper he proved the impossibility of the solution under
  Euclidean restrictions. Wantzel regarded the magnitudes involved not
  as geometric segments, but as numerical lengths, via analytic
  geometry. This let him use algebra and arithmetic rather than pure
  geometry [Dunham, p. 245]." (source)

For details about the proof check out page 7:
http://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/Suzuki.pdf
Ferdinand von Lindemann's proof (1882) that pi is transcendental (non-algebraic and therefore non-constructible) was also important,
because squaring the circle by compass and straightedge would require constructing the square root of pi.
A useful chart: 
http://en.wikipedia.org/wiki/Constructible_number#Impossible_constructions 
A useful explanation:
http://www.uwgb.edu/dutchs/PSEUDOSC/trisect.HTM
History of the three:  
http://www-history.mcs.st-and.ac.uk/Indexes/Greeks.html 
http://www.docstoc.com/docs/73739647/History-topic--Squaring-the-circle
