How can one find a basis of the n-th cyclotomic field as a vector space over Q in a standard way ?
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When proving it is a finite dimensional $\mathbb{Q}$ vector space, you have shown that $\mathbb{Q}(\zeta_n) \simeq \mathbb{Q}[x] / (\Phi_n) \simeq {\sum_{k=0}^{n-1} c_k \zeta_n^{,k} , c_k \in \mathbb{Q}}$ where $\Phi_n$ is the minimal polynomial of $\zeta_n$ (the cyclotomic polynomial) – reuns Nov 08 '16 at 12:45
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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 $[\mathbb{Q}(\zeta_n):\mathbb{Q}]=\varphi(n)$.
Zev Chonoles
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