Cyclotomic Polynomial has Integer Coefficients

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Theorem

Let $n \in \Z_{>0}$ be a positive integer.

Then the $n$th cyclotomic polynomial $\map {\Phi_n} x$ has integer coefficients.


Proof

We proceed by induction on $n$.

For $n = 1$, it follows from First Cyclotomic Polynomial that $\map {\Phi_1} x = x - 1$.


Suppose it is true for all $m<n$.

From Product of Cyclotomic Polynomials:

$\displaystyle \prod_{d \mathop \divides n} \map {\Phi_d} x = x^n - 1$


By the induction hypothesis:

$\displaystyle \prod_{\substack {d \mathop \divides n \\ d \mathop \ne n}} \map {\Phi_d} x$

is a monic polynomial with integer coefficients, and thus primitive.


From Content of Polynomial is Multiplicative it follows that $\map {\Phi_n} x$ has integer coefficients.



$\blacksquare$