For $a=\cos(2\pi/n)$, show that $ = \ldots$

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.

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