# Irreducibility of a polynomial

For $a,b_{1},b_{2} \dots b_{n} \in \mathbb{Z}$, $a>0$ and $b_{i}<b_{i+1}$, 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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Why do you wite $p/{p^n}$ and not just $1/p^{n-1}$? –  Hagen von Eitzen Dec 9 '12 at 21:51
Its the way it is written in Langs book (where I got the problem). Not sure why we wrote it that way –  Digital Gal Dec 9 '12 at 21:54
And there was a typo as well. Sorry –  Digital Gal Dec 9 '12 at 22:02

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

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Thanks. That is a nice trick. –  Digital Gal Dec 9 '12 at 22:31