Proposition: If the constant term is $1$ or $-1$, then we can't use the Eisenstein criterion to determine whether the polynomial is irreducible over $Q$.
Is it right?
Edit
Since directly use is not right. So the proposition is right or not right?
And further question is, When should I try some substitutions to use Eisenstein criterion? Any guidelines and rules?
For example，I found two examples：
$x^7+7x+1$
$\endgroup$ 62 Answers
$\begingroup$The question has already some answer here: Irreducibility check for polynomials not satisfying Eisenstein Criterion.In addition, there are some results on how many monic polynomials in $\mathbb{Z}[x]$ can be shown to be irreducible by Eisenstein's crtiterion. For example, less that $1\%$ of the polynomials with at least seven non-zero coefficients are irreducible by Eisenstein (A. Dubickas, 2003).
Edit: To the new question. No, if the constant term is $\pm 1$, we cannot apply Eisenstein directly, and also not in general after a substitution $x\mapsto x+a$ (e.g., $x^3+x+1$). And yes, there are certain rules, called Eisenstein shifts, when you can attempt a substitution. See the article "On shifted Eisenstein polynomials" of R Heyman, I. Shparlinski (2013).
$\endgroup$ $\begingroup$As noted in the comments, you can't use Eisenstein directly, but you might be able to after a substitution.
$\endgroup$ 2
