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Prove that there are infinitely many values of α for which $ x^{7} +15x^{2} − 30x + α $ is irreducible in $Q[x]$. I know that an irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. I need a general way to show irreducibility. Thanks.

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  • $\begingroup$ sorry, corrected $\endgroup$ – AMT Oct 20 '17 at 11:09
  • $\begingroup$ Fortunately, my answer is also true for this polynomial :) $\endgroup$ – Peter Oct 20 '17 at 11:14
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You can show with the Eisenstein-criterion that every $\alpha$ divisible by $5$ , but not divisible by $25$, does the job.

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