I'm having trouble with the following Lemma:
$\Phi_m$ is defined over the prime subfield of $K$ (that is, over $\mathbb Q$ or $\mathbb F_p$). When $\mathrm{char}K = 0$, $\Phi_m$ is defined over $\mathbb Z$.
The proof proceeds as follows:
The proof is by induction on $m$. The result is trivial if $m = 1$. If $m > 1 $ then $X^m - 1 = \Phi_m \displaystyle \prod _{d|m, d \neq m} \Phi_d = \Phi_m g$, where $g$ is monic and by the induction hypothesis is defined over the prime subfield of $K$ (and over $\mathbb Z$ if $\mathrm{char}K = 0$). By Gauss' Lemma, or by direct argument using the Remainder Theorem, $\Phi_m$ is also defined over the prime subfield (and over $\mathbb Z$ if $\mathrm{char}K = 0$).
I don't understand this at all. How is Gauss' Lemma being used? I'm also confused about the wording, in particular the characteristic $0$ bit. Any field with characteristic $0$ has $\mathbb Q$ as its prime subfield, right? So doesn't that make the $\mathbb Q$ bit redundant? Obviously if the polynomial is defined over $\mathbb Q$ then it's also defined over $\mathbb Z$, since the coefficients are all algebraic integers.