I observed that different books mention properties of cyclotomic polynomials in terms of lemmas (not obvious proof) and some books mention same thing as obvious thing. Here are two of them. Can one point out, what is obvious part and what is non-trivial part, if any, as far as undergraduate student is considered matured with Gallian's book on Algebra?
If $\zeta_n=e^{2\pi i/n}\in\mathbb{C}$, then $$x^n-1=\prod_{1\le i<n} (x-\zeta_n^i). \,\,\,\,\,\,\,\,\,\,(*)$$ Define $n$-th cyclotomic polynomial: $$\Phi_n(x)=\prod_{1\le i<n, (i,n)=1}(x-\zeta_n^i).$$
- Property $1$: $x^n-1=\prod_{d|n}\Phi_d(x)$.
Now Lang in Algebra book mentions Property $1$ by reason Collect all the terms in $(*)$ belonging to roots of unity having the same period. Whereas, David Cox mentions Property $1$ as a proposition, with some non-obvious arguments. (What is obvious and what is non-obvious in proof of property 1?)
- Property $2$: $\Phi_n(x)=\prod_{d|n} (x^d-1)^{\mu(n/d)}$.
Now Gennady Bachman in his book (On the Coefficients of Cyclotomic Polynomials) mentions An immediate consequence of Property 2 is that $\Phi_n(x)$ has integer coefficients; but, most of the books, which prove $\Phi_n(x)\in\mathbb{Z}[x]$ give some arguments involving Gauss lemma. I do not get, how it is obvious from Property $2$ that $\Phi_n(x)$ has integer coefficients? $\mu(n/d)$ can be $-1$ for some $d$.