Extension Field of $e^{\frac{i\pi}{5}}$

Question

Let $w = \cos\frac{2\pi}{10}+ i \sin\frac{2\pi}{10}$.

Let $K=\mathbb Q(w^2)$ and $L=\mathbb Q(w)$. Then

1. $[L:\mathbb Q] = 10$

2. $[L:K] = 2$

3. $[K:\mathbb Q] = 4$

4. $L = K$

Now, once we know the value of $w$, we can find the degree of the extension field, but how to find the value of $w$? or can we find the degrees without finding the $w$ value? please explain..

Answer 1

$w=e^{i\pi /5}$ and thus $X^5-1$ is an annihilator polynomial (you don't need anything else !). Since $$X^5-1=(X-1)(X^4+X^3+X^2+X+1)$$ and that $X^4+X^3+X^2+X+1$ is irreducible (why ?) $$[L:\mathbb Q]=4.$$

For similar reason, $[K:\mathbb Q]=4$, and thus $K=L$ (why ?).