Eisenstein Criterion Shift Conditions I have been working through my Abstract Algebra book and have come across Eisenstein's criterion.  Many of the problems have been proving polynomials are irreducible using a shift in Eisenstein's criterion.  That is, give some $f(x)$, let $f(x+b)=g(x)$ for which we can use Eisenstein's criterion on $g(x)$ to prove that $f(x)$ is irreducible.
I understand the conditions and the proof. However, 

I was wondering if any irreducible polynomial $f(x)$ can be shifted to $g(x)=f(ax+b)$ then use Eisenstein's criterion on $g(x)$?

I was researching some of this and came across the following:
Let $f(x)$ be monic with integer coefficients and let $g(x)=f(ax+b)$ for integers $a$ and $b$. Suppose there exists a prime $p$ for which Eisenstein applies to $g(x)$. Then $f(x)$ is irreducible. Now show that such an $a$ and $b$ exists if and only if $g(x)=c(x-b)^n\pmod p$ for some $c$. The hint was to use Taylor expansions. I have been working on this problem for a while and just cannot seem to figure out how to go about proving the if and only if statement.
Please provide hints. Not answers. Thanks!
 A: Good question! The answer is no, but it's surprisingly hard to produce either an explanation why or a counterexample without introducing some algebraic number theory. 
The short story, which is still not so short, is that if $f(x)$ is, say, a monic integer polynomial of degree $n$, we can associate to it an integer $\Delta$ called the discriminant of the number field it generates, and if Eisenstein's criterion works for a prime $p$ on $f(x)$ or any translate of it (we can set $a = 1$ without loss of generality), then $p^{n-1}$ must divide $\Delta$, and it's possible to write down examples of cubic polynomials $f(x)$ (so that $n-1 = 2$) such that $\Delta$ is squarefree. 
The long story involves a concept in algebraic number theory called ramification. See this PDF for some details, although you might need to crack open a textbook on algebraic number theory first. 
In particular, from Wikipedia I learn that if $f(x) = x^3 - x^2 - 2x - 8$ then the discriminant is $-503$, which is the negative of a prime, so Eisenstein's criterion cannot be used on any translate of $f$. 
