Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 6 down vote accepted

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

share|cite|improve this answer
Thanks. That is a nice trick. – Digital Gal Dec 9 '12 at 22:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.