Cyclotomic Polynomial has Integer Coefficients

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 $n=1$, this is true because the first cyclotomic polynomial is $x-1$.

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 Gauss's Lemma (Polynomial Theory)/Corollary 1 it follows that $\Phi_n(x)$ has integer coefficients.