question on testing irreducibility of a polynomial

Question

currently reading a textbook which states the following - if a polynomial $f$ is reducible in $\mathbb{Z}$, so long $n$ does not divide the highest coefficient of $f$ it is irreducible in $\mathbb{Z}_n$. don't they mean the opposite? if that is the case, why can $n$ not divide the highest coeff of $f$? is it because it will have a lower degree in $\mathbb{Z}_n$?

Answer 1

It should say that if the polynomial is reducible over $\mathbb{Z}$ then it is reducible over $\mathbb{Z}_n$, if the lead coefficient is not divisible by $n$. The ordinary use of this result is to prove irreducibility over $\mathbb{Z}$ by showing irreducibility over a well-chosen $\mathbb{Z}_n$. Maybe that is the reason for the verbal slip.

The reason for the restriction is essentially the one you gave. If the lead coefficient is divisible by $n$, then a splitting of $P(x)$ as a product of polynomials of lower degree may not induce such a splitting when we reduce modulo $n$. As an extreme example, let $n=2$, and consider the polynomial $4x^2-1$.