Let $\displaystyle L$ be the Galois closure of $\displaystyle K=\mathbb{Q}(\sqrt[n]{a})$, where $\displaystyle a \in \mathbb{Q}, a>0$ and suppose $\displaystyle [K:\mathbb{Q}]=n$.

Prove that $\displaystyle [L:\mathbb{Q}]=n \varphi(n)$ or $\displaystyle \frac{1}{2}n\varphi(n)$

($\displaystyle \varphi$ Euler's function)