Cyclotomic Polynomial has Integer Coefficients

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Theorem

Let $n>0$ be a positive integer.

Then the $n$th cyclotomic polynomial $\Phi_n(x)$ has integer coefficients.


Proof

We proceed by induction on $n$.

For $n=1$, it follows from First Cyclotomic Polynomial that $\Phi_1(x)=x-1$.

Suppose it is true for all $m<n$. From Product of Cyclotomic Polynomials we have

$\displaystyle \prod_{d \mathop \backslash n} \Phi_d \left({x}\right) = x^n-1$

By the induction hypothesis,

$\displaystyle \prod_{\substack{d \mathop \backslash n\\ d\neq n}} \Phi_d \left({x}\right)$

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

From Content of Polynomials is Multiplicative it follows that $\Phi_n(x)$ has integer coefficients.

$\blacksquare$