Let $p\neq 2$ be a prime number and $\zeta_p = e^{2\pi i/p}$ be a $p$-th root of unity. Consider the field $K = \mathbb{Q}(\zeta_p + \zeta_p^{-1})$. a) Find a $\mathbb{Q}$-basis for the field $K$. b) Is $\zeta_p \in K$? Find the degree of $\zeta_p$ over $K$.