Let be a polynomial with integer coefficients and assume that for a given prime integer that:

1. does not divide ,

2. divides

3. does not divide

Prove that either has a rational root, or is irreducible over the filed

What is the meaning of the last sentence? Do I have to show that there are two cases: case1 irreducible, case2 has rational root?

I assume that I should consider two cases. Case1 is easy, it occurs when does not divide . In this case Eisenstein's criterion gives answer. But what about case2 when divides ?

I managed to show the following in case2:

where is irreducible according to Eisenstein's criterion. I have a feeling that has a rational root, but do not know how to show it...