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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!

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2 Answers 2

up vote 3 down vote accepted

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.

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Nice work! loved it –  Sean Jul 20 at 18:19

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

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$x^3 + x^2 + x + 1 = (x^2+1)(x+1).$ –  jspecter Jul 20 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 at 18:07

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