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For $a,b_{1},b_{2} \dots b_{n} \in \mathbb{Z}$, $a>0$ and $b_{i}, I'm trying to show the following polynomial is irreducible in $\mathbb Q[x]$:

$f(x)=(x^{2}-a)(x-b_{1})(x-b_{2}) \dots (x-b_{n}) + \frac{p}{p^{n+2}}$

where $p$ is prime. I first tried the case $n=1$ and multipled by $p^{3}$. The only condition I know to check for is Eisenstein's criterion which doesn't seem apply here.

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    And there was a typo as well. Sorry2012-12-09

1 Answers 1

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$p^{n+2}f(x/p)=(x^2-ap^2)(x-pb_1)\cdots(x-pb_n)+p$matches the Eisenstein criterion.