Compass and straightedge constructions

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 ?

-
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

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 $ab+cd$?

-

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.

-