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!
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 $f$.