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I just came across the following interesting question which has been once discussed:

Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree

I was wondering if we could find such irreducible polynomials, I mean for every degree n, none of which satisfying the Eisenstein's Criterion's hypothesis.

Thanks so much in advance!

Community
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Sean
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2 Answers2

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$f(x) = x^{p-1} + x^{p-2} + \ldots + x + 1$ works for every prime $p$.

This is an example of a polynomial that doesn't satisfy Eisenstein's criterion but you can use Eisenstein to show it's irreducible:

$f(x+1) = x^{p-1} +px^{p-2} + \ldots + px + p$

is irreducible by Eisenstein, and hence so is $f(x)$

Mathmo123
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  • $x^3 + x^2 + x + 1 = (x^2+1)(x+1).$ – jspecter Jul 20 '14 at 18:02
  • Corrected... and whilst it now doesn't fully answer the question, I think the answer still has value, since it illustrates how Eisenstein can be used to prove polynomials are irreducible, even when they don't satisfy the criteria – Mathmo123 Jul 20 '14 at 18:07
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Sure. Let $E$ be your favorite monic irreducible polynomial of degree d. Let $c = E(0).$ Then $$ \tilde{E} := c^dE(c^{-1}X) $$ is a monic irreducible polynomial of degree $d$ with constant term $c^{d+1}.$ In paricular $\tilde{E}$ is not Eisenstein.

jspecter
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