irreducibility of $m$th cyclotomic polynomial in $\mathbb Z[x]$ implies irreducibility in $\mathbb Q[x]$? (Ireland and Rosen)

Question

In Chapter 13 of Ireland and Rosen's An Introduction to Classical Modern Algebra, they prove that the $m$th cyclotomic polynomial is irreducible in $\mathbb Z[x]$. Immediately afterwards they state a corollary which is dependent upon the same polynomial being irreducible in $\mathbb Q[x]$. I know that the cyclotomic polynomial is irreducible in $\mathbb Q[x]$ but the fact that Ireland and Rosen's book didn't explicitly use that fact grabs my attention.

Does irredubility in $\mathbb Z[x]$ of the cyclotomic polynomial imply the same for $\mathbb Q[x]$?

Answer 1

The magic words are: Gauss Lemma