Show $\cos(\pi/11)$ is algebraic over $\mathbb{Q}$

Question

Show $\cos(\pi/11)$ is algebraic over $\mathbb{Q}$

I am trying to follow the answers in How to prove that $\cos (2\pi/n)$ is algebraic?

So $\cos(\pi/11)+ i\sin(\pi/11)$ is a root of $x^{22}=1$ and so $\cos(\pi/11)+ i\sin(\pi/11)$ is algebraic. Since, sums and scalar multiples are algebraic then $\cos(\pi/11)$ is algebraic. Is that right?

Answer 1

I am unable to follow that argument. But $\cos\left(\frac\pi{11}\right)+i\sin\left(\frac\pi{11}\right)$ is indeed a root of $x^{22}-1$. And so is $\cos\left(\frac\pi{11}\right)-i\sin\left(\frac\pi{11}\right)$. But then their sum, which is $2\cos\left(\frac\pi{11}\right)$, is algebraic. So, $\cos\left(\frac\pi{11}\right)$ is algebraic.