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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 $\mathbb Q[x]$?

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Gauss' Lemma: a primitive polynomial in $\mathbb{Z}[X]$ is irreducible if and only if it is irreducible in $\mathbb{Q}[X]$. – egreg Dec 5 '13 at 22:47
up vote 4 down vote accepted

The magic words are: Gauss Lemma

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