Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

Question

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!

Answer 1

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.

Answer 2

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