Math Help - Basis of primitive nth roots of unity when n is squarefree?

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 th roots of unity in the th cyclotomic field form a normal basis over if and only if is squarefree.

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

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 th roots of unity in the th cyclotomic field form a normal basis over if and only if is squarefree.

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

do it by induction over . first assume that is a prime and solve the problem. then write where is a prime and is an integer coprime with and apply your induction hypothesis.

Thanks NonCommAlg. I understand how it follows when is prime, since the powers of the adjoined element in a simple extension form a basis. But when , is squarefree, so by the induction hypothesis, the primitive th roots of unity form a basis for . By the result of simple extensions, the primitive th roots of unity form a basis of over , so letting the basis run over products of primitive roots of and gives a basis of primitive th roots of over ? Is that the correct way to go?