I'm solving problems from my galois theory course to practice for my exam and I found this one:
Calculate the degree of the minimal polinomial of $\cos(2\pi/p)$ over $\mathbb{Q}$, where $p$ is a prime number. Hint: compare the appropiate extension to $\mathbb{Q}(e^{i2\pi/p})/\mathbb{Q}$.
I followed the hint and found that $[\mathbb{Q}(e^{i2\pi/p}):\mathbb{Q}]=p$ (correct me if I'm wrong), because given that $p$ is prime, $\{1,e^{i2\pi/p},e^{i2\pi\cdot2/p},\dots, e^{i2\pi(p-1)/p}\}$ is $\mathbb{Q}$-linearly independent (thought I'm not pretty sure if this is trivial or how can I prove it anyway), while $\{1,e^{i2\pi/p},e^{i2\pi\cdot2/p},\dots, e^{i2\pi(p-1)/p},e^{i2\pi p/p}=1\}$ is $\mathbb{Q}$-linearly dependent, so this gives the result that $[\mathbb{Q}(e^{i2\pi/p}):\mathbb{Q}]=p$.
Now, if I prove $\cos(2\pi/p)\in\mathbb{Q}(e^{i2\pi/p})$, for generation minimality I conclude $\mathbb{Q}(\cos(2\pi/p))\subset \mathbb{Q}(e^{i2\pi/p})$ and can use degree transitivity to find that $[\mathbb{Q}(\cos(2\pi/p)):\mathbb{Q}]$ is either $p$ or $1$ (I don't know yet which of this is the case). I'm stuck here, proving $\cos(2\pi/p)\in\mathbb{Q}(e^{i2\pi/p})$ and then finding if either $\mathbb{Q}(\cos(2\pi/p))=\mathbb{Q}(e^{i2\pi/p})$ or either $\mathbb{Q})\cos(2\pi/p))=\mathbb{Q}$.
How can I end this? Is my reasoning till this point correct? If not, please correct me. Any help will be appreciated, thanks in advance.
You may try denoting $e^{i2\pi/p}=a$, and notice that $\cos(2\pi/p)=\frac{a+a^{-1}}{2}$.