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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：

$x^7+7x+1$

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forlorn
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  • Eisenstein's criterion requires that the constant term be a multiple of some prime number $p$. The numbers $\pm 1$ are the only ones that do not have a prime factor. – lhf Aug 08 '13 at 10:43
  • Yes, if the constant term is $1$ then we cannot use Eisenstein directly to prove irreducibility, since there is no prime we could apply it with. – Tobias Kildetoft Aug 08 '13 at 10:44
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    However, as the wikipedia link @lhf put forward points out under Examples - Cyclotomic polynomials, substituting $x$ for $(x-1)$ or $(x+1)$ and expanding the brackets, might just make a suitable prime pop out on the other end. Any other substitute $(x+a)$ for an integer $a$ might also work, so while it might be long and tedious to calculate, it might just be worth it in the end. – Arthur Aug 08 '13 at 10:57
  • Sometimes (but rarely) a substitution helps, e.g. $f(x)=x^4+4x^3+10x^2+12x+7$ becomes, after $x\mapsto x+1$, $g(x)=x^4+4x^2+2$, which is irreducible by Eisenstein. – Dietrich Burde Aug 08 '13 at 11:26
  • @DietrichBurde When can I try the substitution, is there any rules? – forlorn Aug 08 '13 at 11:58
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    Yes, for example with $x^3+x+1$ a substitution is useless, see the answer in http://math.stackexchange.com/questions/458802/irreducibility-check-for-polynomials-not-satisfying-eisenstein-criterion. – Dietrich Burde Aug 08 '13 at 12:00

2 Answers2

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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).

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Dietrich Burde
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As noted in the comments, you can't use Eisenstein directly, but you might be able to after a substitution.

Gerry Myerson
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  • Yes, but the substitutions you might perform are quite limited, see http://math.stackexchange.com/questions/458802/irreducibility-check-for-polynomials-not-satisfying-eisenstein-criterion. – Dietrich Burde Aug 08 '13 at 11:24
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    When can I try the substitution, is there any rules? – forlorn Aug 08 '13 at 11:58