For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$.
$\phi$ is the Euler totient function which gives the number of coprime elements.
We are new to abstract algebra. This is a question on a difficult project involving cyclotomic polynomials and their irreducibility.
We know that the degree of the cyclotomic extension is just $\phi(n)$, but obviously $a$ is just one piece of the roots of unity.