Show that $[L: K]=2$, where $L=\{p+q\zeta +r\zeta^2+s\zeta^3: p, q, r, s \in \mathbb{Q}\}$, $K=\{a+b(\zeta^2+\zeta^3): a, b \in \mathbb{Q}\}$, and $\zeta=e^{\frac{2\pi i}{5}}$.
The conjecture is an intermediate step for another problem.
Attempt: Clearly I want to show that $L=K(\sqrt{\alpha})$ for some $\alpha \in K$; more specifically, $\zeta=\sqrt{\alpha}$. But I struggle to come up with an $\alpha$ that works. Is the conjecture false, or am I incompetent?
Any hint would be greatly appreciated.
Clearly, $X^4+X^3+X^2+X+1$ is the minimal polynomial of $\zeta$ over $\mathbb{Q}$. So, $[L:\mathbb{Q}] = 4 = [L:K][K:\mathbb{Q}]$. Note that $\mathbb{Q} \subsetneq K \subsetneq L$, since $\zeta^3+\zeta^2 = \bar\zeta^2+\zeta^2 = -\frac{1}{2}-\frac{\sqrt{5}}{2} \in \mathbb{R} \setminus \mathbb{Q}$ and $L \nsubseteq \mathbb{R}$. This means $[L:K]=[K:\mathbb{Q}] = 2$, since $4>[K:\mathbb{Q}] > 1$ and $[L:\mathbb{Q}] = 4 = [L:K][K:\mathbb{Q}]$.