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I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.

For instance, for the family of polynomials $$ p_n(x)=x^n-x^{n-1}-1 $$ Wolfram Alpha suggests that $p_n$ is probably irreducible for $n\ge6$. How does one show this?

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The book by Prasolov "Polynomials" has lot of information that you seek.

To give one specific instance take the infinite series of $e^x$ truncate it. Even though it is not in the form you desire, it is an irreducible polynomial. This result is due to Schur and is very old. This has been generalized, and it may be in the form you need it.

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