Suppose is reducible. Then, has a solution. By reducing the modulus, we have , so is divisible by . Write . Putting this back into the original equation gives , which is certainly not true.
Therefore, is irreducible.
Consider the polynomial modulo .
I would like to show that this polynomial is irreducible. Unfortunately I don't see how I can apply any of the usual tricks (Eisenstein, simple enumeration in the case of fields of small characteristic) might work.