6

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.

Mykie
  • 7,037

1 Answers1

7

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

Hagen von Eitzen
  • 374,180