Originally Posted by

**AshleyLin** I found an unanswered question on Math.SE concerning a footnote in an expository paper by Keith Conrad. It states that in general, the primitive $\displaystyle n$th roots of unity in the $\displaystyle n$th cyclotomic field form a normal basis over $\displaystyle \mathbf{Q}$ if and only if $\displaystyle n$ is squarefree.

The forward direction is not difficult to show. However, if $\displaystyle n$ is squarefree, then how can one show that the primitive $\displaystyle n$th roots of unity form a basis for the cyclotomic extension over $\displaystyle \mathbb{Q}$?