Eisenstein's criterion states that for a polynomial
$$f(X)=a_0 + a_1X + ... + a_nX^n$$
with $a_0, ..., a_n \in \mathbb{Z}$ (or more generally a UFD), then if there exists a prime $p$ such that
- $p \nmid a_n$
- $p \mid a_i$, $i < n$
- $p^2 \nmid a_0$
then it is irreducible over $\mathbb{Q}$ (more generally, the field of fractions of the UFD).
Eisenstein's criterion may not apply for some polynomial $f(X)$, but applies for $f(X+c)$ for some $c \in \mathbb{Z}$. An example (from this Wikipedia article) is as follows:
$$f(X)=X^2+X+2$$
does not satisfy Eisenstein's criterion, but
$$f(X+3)=X^2+7X+14$$
does, with $p=7$.
We know that $f(X)$ irreducible $\iff$ $f(X+c)$ irreducible (if one were reducible, you could substitute the transformation into the factors to get the other is reducible.) Thus, this "translation trick" can be useful.
My question is if there is any way to whittle down the possible values of $c$ we need to check to confirm it is irreducible, or that possibly work? What would be particularly useful is if there was some kind of bound on $|c|$, so there's only finitely many cases to check. Of course, if no values of $c$ work, this may not necessarily prove it is reducible, as far as I am aware.
I'm struggling to find any result that provides any more intuition on what values of $c$ to try, other than just guess-and-check - perhaps there is no known way. But if someone with better knowledge (or better Googling skills!) than me could fill in the gaps, that would be much appreciated.
