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Is there elementary proof that, for example, $\sqrt[3]2$, cannot be constructed using rule and compass?

Also, my guess is that only roots which are powers of $2$ (2nd root, 4th root, 8th root, etc.) can be geometrically constructed, is this true?

Please, do not use higher mathematics such as Galois theory because I have only 1st year university math knowledge.

1b3b
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    Your guess is right, but I doubt that there is a proof using nothing from Galois theory. – Peter Jul 03 '20 at 11:37
  • Oh, nice (guess). Can you recommend some literature for learning Galois theory for beginners? – 1b3b Jul 03 '20 at 11:38
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    There is a reason it took until the 19th century to prove you can't double the cube. If there were a more elementary proof, chances are high the ancient Greeks, or any of a number of really clever mathematicians who attempted a proof in the meantime, would've found it. – Arthur Jul 03 '20 at 11:39
  • @Arthur, thanks. Honestly, I did not search any history of this problems, just the proof. – 1b3b Jul 03 '20 at 11:43
  • See also https://math.stackexchange.com/a/53648/589 – lhf Jul 03 '20 at 12:54
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    Look at this book - https://indianaeducations.files.wordpress.com/2016/01/famous-problems-of-geometry-how-to-solve-them-benjamin-bold.pdf – Moti Jul 03 '20 at 16:18

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