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As opposed to the generic polynomial form for utilizing the Eisenstein Criterion ($a_nx^n+a_{n-1}x^{n-1}+\dots+a_0\in\mathbb{Z}[x]$ is irreducible in $\mathbb{Q}$) how do we prove that if $p$ is a prime, $x^{p-1}+x^{p-2}+\dots+x+1$ is irreducible over $\mathbb{Q}$?

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    Let $x=y+1$ and plug in; if the resulting polynomial is irreducible, then so is the original. By the way, it's either "irreducible in $\mathbb{Q}[x]$", or "irreducible **over** $\mathbb{Q}$".2011-12-02
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    See also [Irreducibility of $X^{p-1} + \ldots + X+1$](http://math.stackexchange.com/questions/215042/irreducibility-of-xp-1-ldots-x1)2013-06-01

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