How can I show that $X^p-X-1\in \mathbb Q[X]$ is irreducible ? I tried reduction modulo $p$, but doesn't work, even some normal variable changing as $X\longmapsto X+1$ or $X\longmapsto X-1$, but nothing. Any idea ?
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2Reduction mod $p$ does work. But you also need this perfectly fitting result. – Jyrki Lahtonen Jan 04 '16 at 20:28
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Possible duplicate of Show that $f(x) = x^p -x -1 \in \Bbb{F}_p[x]$ is irreducible over $\Bbb{F}_p$ for every $p$. – Dietrich Burde Jan 09 '19 at 10:26