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

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    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 nth roots of unity in the nth cyclotomic field form a normal basis over \mathbf{Q} if and only if n is squarefree.

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

    Quote Originally Posted by AshleyLin View Post
    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 nth roots of unity in the nth cyclotomic field form a normal basis over \mathbf{Q} if and only if n is squarefree.

    The forward direction is not difficult to show. However, if n is squarefree, then how can one show that the primitive nth roots of unity form a basis for the cyclotomic extension over \mathbb{Q}?
    do it by induction over n. first assume that n is a prime and solve the problem. then write n = kp, where p is a prime and k < n is an integer coprime with p, and apply your induction hypothesis.
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    Re: Basis of primitive nth roots of unity when n is squarefree?

    Thanks NonCommAlg. I understand how it follows when n is prime, since the powers of the adjoined element in a simple extension form a basis. But when n=kp, k is squarefree, so by the induction hypothesis, the primitive kth roots of unity form a basis for \mathbb{Q}(\zeta_k)/\mathbb{Q}. By the result of simple extensions, the primitive pth roots of unity form a \mathbb{Q}(\zeta_k) basis of \mathbb{Q}(\zeta_k,\zeta_p) over \mathbb{Q}(\zeta_k), so letting the basis run over products of primitive roots of \zeta_p and \zeta_k gives a basis of primitive nth roots of \mathbb{Q}(\zeta_k,\zeta_p) over \mathbb{Q}? Is that the correct way to go?
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