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I'm studying field theory and I was given an exercise:

Is $\sqrt[3]2\cos\frac{2\pi}{34}+\sqrt5\cos\frac{2\pi}{10}$ a constructible point ?

Any hints ?

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btw: How can I do a third root notation in tex ? – Belgi Apr 19 '12 at 10:44
$\sqrt[3]{}$ is written as \sqrt[3] in TeX! – user21436 Apr 19 '12 at 10:47
Or \root3\of{}. – Gerry Myerson Apr 19 '12 at 12:44
A comprehensive book on ruler/compass and other geometric construction techniques and theory is available in the excellent book by George Martin: Geometric Constructions, Springer, 1998. – Joseph Malkevitch Apr 19 '12 at 15:02
up vote 7 down vote accepted

Hint: If you're ready to solve this exercise, you should already know the answers to:

  1. Is $a=\sqrt[3]{2}$ constructible?
  2. Is $b=\cos\frac{2\pi}{34}$ constructible?
  3. Is $c=\sqrt{5}$ constructible?
  4. Is $d=\cos\frac{2\pi}{10}$ constructible?

What do the anwers to these four questions imply about the constructibility of $ab+cd$?

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Here is a 5-step proof:

  1. Show that $\cos \frac{1}{5} \pi$ and $\cos \frac{1}{17} \pi$ are both constructible.

  2. Show that $\sqrt{5}$ is constructible.

  3. Show that the sum and product of two constructible numbers are constructible, and show that the additive and multiplicative inverses of a (non-zero) constructible number are constructible.

  4. Show that $\sqrt[3]{2}$ is not constructible.

  5. Suppose $\sqrt[3]{2} \cos \frac{1}{17} \pi + \sqrt{5} \cos \frac{1}{5} \pi$ is constructible and deduce a contradiction.

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