Proving that cyclotomic polynomials have integer coefficients

Here are some remarks :

• For those who haven't Dummit and Foote (lemma 40, Sec. 13.6 in the third edition), let me recall that they define $\Phi_n(X)$ as the product of the $X-\zeta$ with $\zeta \in \mu_n$ going through the primitive $n$-th roots of unity.

• In their proof, they say : $f$ divides $X^n-1$ in $\Bbb Q(\zeta_n)[X]$ and also in $\Bbb Q[X]$ by the division algorithm. Then, $X^n-1=f(X)\Phi_n(X)$ in $\Bbb Q[X]$ and thanks to Gauss' lemma ($f$ being monic in $\Bbb Z[X]$), one has $f \in \Bbb Z[X]$ and $\Phi_n \in \Bbb Z[X]$. In this version, they need to use Gauss' lemma.

• But I think your version is also right. Actually, since $f$ is assumed to be a monic polynomial in $\Bbb Z[X]$, the division algorithm allows us to write $X^n-1 = q(X)f(X)+r(X)$ in $\Bbb Z[X]$ as you did. In particular, you don't need to use Gauss' lemma here. Then the conclusion follows by your argument, which seems correct to me.

• Sometimes, the $n$-th cyclotomic polynomial is defined as the minimal polynomial of $\zeta_n$ over $\mathbb Q$. Then $\Phi_n$ divides $X^n-1$ in $\mathbb Q[X]$, since it is a minimal polynomial. By Gauss' lemma, we get : $\Phi_n$ divides $X^n-1$ in $\mathbb Z[X]$, in particular $\Phi_n \in \Bbb Z[X]$.