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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.

user26857
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user148884
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    If $\zeta = e^{2 \pi i / n}$, can you think of a way to express $a$ in terms of $\zeta$? Remembering the complex definition of cosine might be useful here... – bzc Jun 06 '14 at 05:55
  • We are trying to use that, essentially, a = [e^((2(pi)i)/n)+(e^((2(pi)i/n)))^(n-1)]/2 and tried to find something that way. We also have tried to just simply understand why the degree is equal to half the number of coprime elements to our choice of n. It works just though examples but we can't expand it to a general case. – user148884 Jun 06 '14 at 06:16
  • Do you know for a fact that $\Bbb [\Bbb Q(\alpha):\Bbb Q] = \phi(n)/2$ is the case or is it a conjecture on your part? – Robert Lewis Jun 06 '14 at 07:09
  • Show that $|{\mathbb Q}(\zeta):{\mathbb Q}| = \phi(n)$ and that $|{\mathbb Q}(\zeta):{\mathbb Q}(a)| = 2$ (where $\zeta = e^{2\pi i/n}$). – Derek Holt Jun 06 '14 at 11:25
  • Related: https://math.stackexchange.com/questions/1273388, https://math.stackexchange.com/questions/464300/, https://math.stackexchange.com/questions/460930/, https://math.stackexchange.com/questions/239316/ – Watson Jan 06 '17 at 10:36

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