How to know equation solution irrational

Question

I would appreciate if somebody could help me with the following problem

Q: Is the solution of the equation $$2\cos^2\pi x+\cos \pi x-2=0 $$ irrational ?

Answer 1

We are considering a value $x$ for which $\cos(\pi x) = \frac{\sqrt{17} - 1}4$.

For a prime $p$ in general we have

$$\sum_{k=1}^{p-1}\left(k\over p\right)\zeta_p^k = \left(-1\over p\right)\sqrt p$$

where $\left(\cdot\over p\right)$ is the Legendre symbol and $\zeta_p$ is a primitive $p$-th root of unity. From this we see that $\cos\pi x \in\mathbb Q(\zeta_{17}).$

The only primitive roots of unity contained in $\mathbb Q(\zeta_{17})$ are the 1st, 2nd, 17th and 34th primitive roots of unity.

Now suppose that $x\in\mathbb Q$, say $x/2 = p/q$ with $(p,q) = 1$. It can be directly checked that $q\ne 1,2,17,34$, so that neither $\zeta_q^p$ and $\zeta_q^{-p}$, which are primitive $q$-th roots of unity, nor their sum are in $\mathbb Q(\zeta_{17})$, so that if $x\in\mathbb Q$

$$\cos\pi x = {\zeta_q^p + \zeta_q^{-p}\over 2}\notin\mathbb Q(\zeta_{17})$$

In particular it cannot be a root of your equation if $x\in\mathbb Q$.