For $a,b_{1},b_{2} \dots b_{n} \in \mathbb{Z}$, $a>0$ and $b_{i} $$f(x)=(x^{2}-a)(x-b_{1})(x-b_{2}) \dots (x-b_{n}) + \frac{p}{p^{n+2}}$$ 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.
Irreducibility of a polynomial
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abstract-algebra
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0Why do you wite $p/{p^n}$ and not just $1/p^{n-1}$? – 2012-12-09
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0Its the way it is written in Langs book (where I got the problem). Not sure why we wrote it that way – 2012-12-09
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0And there was a typo as well. Sorry – 2012-12-09
1 Answers
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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.