What is $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q(\sin(2\pi k/n))$? [closed]
Let $\zeta_n$ be the $n$-th primitive root of unity and $4 \mid n$. Consider the field extensions $\mathbb Q \subset \mathbb Q(\sin(2\pi k/n) \subset \mathbb Q(\zeta_n)$.
- What is the degree of the extension $[\mathbb Q(\zeta_n):\mathbb Q(\sin(2\pi k/n)]$?
- What is $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q(\sin(2\pi k/n))$?
You should first prove that $[\mathbb Q(\cos(2\pi /n)):\mathbb Q] =\phi(n)/2$, which is easy.
Then you should use that $\sin(2\pi k/n)=\cos ((n-4k)\pi/2n)$.
A few calculations necessitating case distinctions will then yield the result:
$$[\mathbb Q(\sin(2\pi /n)):\mathbb Q]=\phi(n) \quad \text {if} \; n \not \equiv 0 \;( \text {mod} \; 4) $$ $$[\mathbb Q(\sin(2\pi /n)):\mathbb Q]=\phi(n)/2 \quad \text {if} \; n \equiv 0 \;( \text {mod} \; 8)$$ $$[\mathbb Q(\sin(2\pi /n)):\mathbb Q]=\phi(n)/4 \quad \text {if} \; n \equiv 4 \;( \text {mod} \; 8)$$ [I have answered your question in the above form because (as Jyrki noted) the original question doesn't make sense, since we don't always have an inclusion $\mathbb Q(\sin(2\pi /n))\subset \mathbb Q(\zeta_n).]$
Since $e^{ix} = \cos(x) + i \sin(x)$ we have $$ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i}. $$ Thus $\sin(2 \pi k/n)$ is, unless $n$ is a multiple of $4$, an element of ${\mathbb Q}(\zeta_{4n})$. The degrees of your extensions are basically measured by the greatest common divisors of $k$ and $n$. There will be clean formulas for prime values of $n$, or e.g. if $k = 1$.