Basis of the n-th cyclotomic field over Q

Question

How can one find a basis of the n-th cyclotomic field as a vector space over Q in a standard way ?

Thank you !

Answer 1

If $[\mathbb{Q}(\alpha):\mathbb{Q}]=r$, then $\{1,\alpha,\ldots,\alpha^{r-1}\}$ is a basis for $\mathbb{Q}(\alpha)$ as a $\mathbb{Q}$-vector space.

Now recall that $[\mathbb{Q}(\zeta_n):\mathbb{Q}]=\varphi(n)$.