I have some issues understanding the statement and precise application domain of Gauss lemma. Here is my problem, which I believe I can solve by appealing to it: let $f, g, h$ be monic polynomials of $\mathbb{Q}[X]$ such that $fgh = X^n – 1$. I would like to concluce, since they are monic, that they are all in $\mathbb{Z}[X]$.
However, Gauss lemma states that since $X^n-1$ is reducible over $\mathbb{Q}$, it is also reducible over $\mathbb{Z}$. But can I obviously deduce that the $f, g, h$ of the hypothesis are over $\mathbb{Z}$?
Best Answer
The answer is yes. If $fgh = X^n-1$ with $f,g,h$ monic in $\mathbb Q[X]$ then $f,g,h \in \mathbb Z[X]$. The way to see it is using the usual statement of Gauss Lemma: The product of primitive polynomials is primitive. A primitive polynomial is one such that the gcd of it's coefficients is 1.
Now to see how this implies the assertion find $a,b,c$ minimum positive integers such that $af$, $bg$ and $ch$ have integer coefficients, then they are all primitive primitive so by Gauss Lemma their product $$(af)(bg)(ch) = (abc)(X^n-1) $$ is also primitive, which is obviously false if $abc>1$.