Compass and straightedge constructions

Question

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 ?

Answer 1

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 $ab+cd$?

Answer 2

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.