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.

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    $\begingroup$ A lot of the references in my manuscript A Detailed and Elementary Solution to $x^{17} = 1$ should be useful for what you want. $\endgroup$ – Dave L. Renfro Oct 3 '14 at 18:08
  • $\begingroup$ Squaring the circle is entirely different, unconnected with Galois Theory. The other two do not use Galois Theory proper (Galois group), but come closer. $\endgroup$ – André Nicolas Oct 3 '14 at 18:17
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    $\begingroup$ I should have said Field Theory instead of Galois Theory. $\endgroup$ – jiyanez Oct 3 '14 at 18:27
  • $\begingroup$ Fermat's unknown method can prove doubling the cube by FLT. $\endgroup$ – Takahiro Waki Oct 25 '18 at 12:11

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:

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:




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