In my rings and modules course we've learned some results relating factorizability of polynomials in $\mathbb{Z}[x]$ to factorizability of polynomials in $\mathbb{Q}[x]$ and $\mathbb{F}_p[x]$. Some of these seem unclear to me. For instance, one lemma is stated as follows:
If $f(x)$ is an integer polynomial whose leading coefficient is not divisible by the prime p, and $\overline {f(x)}$ is irreducible in $\mathbb{F}_p[x]$ then $f(x)$ is irreducible in $\mathbb{Z}[z]$.
This seems clearly false to me? For instance, the polynomial $f(x)=2x+6$ has a leading coefficient not divisible by 3, and modulo 3 is the irreducible polynomial $\overline {f(x)}=\overline {2x}$. However, it is reducible in $\mathbb{Z}[x]$, as it factors into $f(x)=2(x+3)$ and neither $2$ nor $x+3$ is a unit.
The lemma is clearly meant to refer only to factorizations into polynomials with positive degree, in which case the proof given in my notes is valid, and so this might seem like a trivial concern, but for instance it also affects Eisenstein's criterion, which for the same reason has an invalid proof as given in my notes. Are these just typos or am I missing something here?
You are confusing irreducibility in $\mathbb{Z}[X]$ with irreducibility in $\mathbb{Q}[X]$. By Gauss's lemma, a polynomial $f(X)\in \mathbb{Z}[X]$ is irreducible in $\mathbb{Z}[X]$ if and only if $f(X)$ is irreducible in $\mathbb{Q}[X]$ and primitive. The two criteria that you mention concern the irreducibility in $\mathbb{Q}[X]$. If you want a statement for the irreducibility in $\mathbb{Z}[X]$, then you need to restrict to $f$ primitive.