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 ?
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 ?
Hint: If you're ready to solve this exercise, you should already know the answers to:
What do the anwers to these four questions imply about the constructibility of $ab+cd$?
Here is a 5-step proof:
Show that $\cos \frac{1}{5} \pi$ and $\cos \frac{1}{17} \pi$ are both constructible.
Show that $\sqrt{5}$ is constructible.
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.
Show that $\sqrt[3]{2}$ is not constructible.
Suppose $\sqrt[3]{2} \cos \frac{1}{17} \pi + \sqrt{5} \cos \frac{1}{5} \pi$ is constructible and deduce a contradiction.