Prove that $\theta$ is an irrational multiple of $2\pi$ given $\cos(\theta/2)\equiv \cos^2(\pi/8)$

Question

How do we prove that $\theta$ is an irrational multiple of $2\pi$ given $\cos(\theta/2)\equiv \cos^2(\pi/8)$?

With $\operatorname{SU}(2)$ rotations,

\begin{align}R_z(\pi/4)R_x(\pi/4)&=[\cos(\pi/8)I-i\sin(\pi/8)Z][\cos(\pi/8)I-i\sin(\pi/8)X]\\ &=\cos^2(\pi/8)I-i[\cos(\pi/8)X+\sin(\pi/8)Y+\cos(\pi/8)Z]\sin(\pi/8)\\ &=\cos(\theta/2)I-i(\hat{n}.\vec{\sigma})\sin(\theta/2)=R_\hat{n}(\theta)\end{align}

where $\vec{n}=(\cos(\pi/8),\sin(\pi/8),\cos(\pi/8))$ and $\hat{n}=\frac{\vec{n}}{||\vec{n}||}$, and $\vec{\sigma}=(X,Y,Z)$ where $X,Y,Z$ are Pauli matrices. Thus $\cos(\theta/2)\equiv\cos^2(\pi/8)$ and $\sin(\theta/2)\equiv\sin(\pi/8)\sqrt{1+\cos^2(\pi/8)}$.

Original Context in my Reference

Ref. to Page 196, 214 of QC and QI by Nelsen and Chuang

Any hint on the possible ways to approach this could be appreciated.

Note : Publication which possibly contains the proof

Answer 1

Let $p,q$ be positive coprime integers and suppose by contradiction that $\theta=2\pi p/q$. Since $\cos^2(\pi/8)=(2+\sqrt2)/4$, it has algebraic degree $2$. By considering Galois conjugates of primitive roots of unity, we can show that over the rationals, the algebraic degree of $\cos(p\pi/q)$ is $\phi(q)$ when $q$ is even, and $\phi(q)/2$ when $q$ is odd.

Therefore, we have $\phi(q)=2$ when $q$ is even, giving $q=4,6$ (when $q$ is odd there are no solutions to $\phi(q)=4$). As none of $\cos(\pi p_1/4)$ or $\cos(\pi p_2/6)$ equals $(2+\sqrt2)/4$ for each integer $p_1\in\{1,3\}$ and $p_2\in\{1,5\}$, we obtain the desired contradiction.