Is $\cos(\frac{2\pi x}{5})$ an element of $\mathbb{Q}(\sqrt{5})$

Question

This question is a little out of context of the full problem, but I'm basically trying to show that $\mathbb{R} \cap \mathbb{Q}[\omega] \subset \mathbb{Q}[\sqrt{5}]$ where $\omega = e^{\frac{2\pi i}{5}}$, the 5th root of unity. I think I see that $\mathbb{R} \cap \mathbb{Q}[\omega]$ is the set of real elements of $\mathbb{Q}[\omega]$, thus I determined the elements are just sums of rational multiples of $\cos(\frac{2\pi x}{5})$ where $0\leq x \leq 4$. So if I show $\cos(\frac{2\pi x}{5})$ is in $\mathbb{Q}[\sqrt{5}]$ then I'm done.

I'm just not so sure how to do this but was thinking it involved how $\cos(\frac{2\pi x}{5})$ is seen when plotted on the unit circle.

Answer 1

You can use explicit formulas for angles in a pentagon (they will involve golden ratio). But it's also possible to get through algebraically. Your original field is defined by roots of unity:

$$z^5-1=0$$ Split (with $z\neq 0$): $$(z-1)(z^4+z^3+z^2+z+1)=z^2(z-1)((z^2+z^{-2})+(z+z^{-1})+1)=0$$ We are looking for real parts of the solutions. Take a look at the last term, writing $w=z+z^{-1}$ and noticing $w^2=z^2+z^{-2}+2$. You get:

$$w^2+w-1=0$$ with solutions $$w=\frac{-1\pm \sqrt{5}}{2}$$

We have $w=z+z^{-1}$ and we know that $z$ lie on the unit circle, so $w=2\operatorname{Re} (z)$, and thus $$\operatorname{Re}(z)\in\mathbb{Q}(\sqrt{5}) $$

Answer 2

$\mathbb{Q}(\sqrt{5})$ is a subfield of $\mathbb{Q}(\omega)$ by Gauss sums. If we consider $$ a = 1+\omega-\omega^2-\omega^3+\omega^4 = 1+2\cos\frac{2\pi}{5}-2\cos\frac{3\pi}{5}$$ we may easily check that $a=\sqrt{5}$ by squaring. On the other hand $$ -\cos\frac{3\pi}{5} = \cos\frac{8\pi}{5} = T_4\left(\cos\frac{2\pi}{5}\right)$$ so $\sqrt{5}\in\mathbb{Q}\left(\cos\frac{2\pi}{5}\right)$, too. By comparing the degrees of the extensions, we get $\mathbb{Q}(\sqrt{5})=\mathbb{Q}\left(\cos\frac{2\pi}{5}\right)$.