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As defined in Wikipedia (and this is the same definition I was given in class), it is not clear to me why the cyclotomic polynomial is over $\mathbb{Q}$.

It is over $\mathbb{C}$, but I don't see a reason for the coefficient to be in $\mathbb{Q}$.

Can anyone help with this one?

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It follows from for example the Möbius inversion formula on that Wikipedia page. –  Jyrki Lahtonen May 9 '12 at 20:25
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Coefficients are in fact in $\mathbb{Z} \subseteq \mathbb{Q}$. –  Sasha May 9 '12 at 20:25
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@Belgi, that application of the Möbius function does not involve complex analysis at all. –  Jyrki Lahtonen May 9 '12 at 20:30
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@Sasha, when you divide two monic polynomials with integer coefficients, the result will also have integer coefficients. Think about what happens when you do long division. –  Jyrki Lahtonen May 9 '12 at 20:32
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1 Answer

up vote 3 down vote accepted

It follows from the following formula and induction on n.

$\prod_{d\mid n}\Phi_d(X) = X^n - 1$

Edit Let $g(X) = \prod_{d\mid n, d < n}\Phi_d(X)$. By the induction hypothesis $g(X) ∈ \mathbb{Q}[X]$. Hence $\Phi_n(X) = (X^n - 1)/g(X) ∈ \mathbb{Q}[X]$

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How does it follow ? –  Belgi May 9 '12 at 20:44
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@Belgi Then you use Jyrki's comment on division of monic polynomials with integer coefficients (and induction) –  M Turgeon May 9 '12 at 20:49
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