Well, my question is essentially:
Let $R$ be a Factorial Ring (UFD, basically) and let $p$ be a prime element in $R$. Let $d$ be an integer larger than 2, and let
$f(t) = t^d + c_{d-1} t^{d-1} + ... + c_0$ be a polynomial belonging to $R[t]$. Let $n \ge 1$ be an integer.
Show that $g(t) = f(t) + \displaystyle \frac{p}{p^{nd}}$ is irreducible in $K[t]$, where $K$ is the quotient field of $R$.
Okay, well, I know that it is enough, from Gauss' Lemma, to show that (after multiplying through by $p^{nd}$) $g$ has no non-trivial factorizations in $R$ itself. EDIT: My apologies, I forgot that this holds specifically for primitive polynomials, and no such specification was given. Nevertheless, perhaps, I suppose the gcd of the coefficients could be factored out?
I also tried thinking of $g$ as essentially just a translation of the graph of $f$ slightly upward, and that $g$ converges to $f$ as $n \to \infty$, but I don't know exactly what to make of that, because, well, as $n$ gets larger, it starts to inch towards the roots of $f$, and would no longer be irreducible, but perhaps I've confused myself.
(As an auxilliary question, there was a previous exercise which asked us to prove that if $R = \mathbb{Z},$ then, if $f$ had $m$ real roots, then, "for sufficiently large $n$", $g$ also has $m$ EDIT: real roots. This should be fairly apparent, I should think, from the graph? But if not, then how would one rigorously establish this?)
Anyway, thanks a lot.