Is $13x^5 + (3 − i)x^3 + (8 − i)(x^2 − x) + 1 − 2i$ irreducible in $(\mathbb{Q}[i])[x]$?

Question

Is $$13x^5 + (3 − i)x^3 + (8 − i)(x^2 − x) + 1 − 2i$$ irreducible in $(\mathbb{Q}[i])[x]$?

I've tried using Eisenstein’s irreducibility criterion to prove that it is, but I don't think it applies here. Would anyone be able to tell me how to approach this?

Answer 1

Since the Gaussian integers are a UFD, the Eisenstein criterion carries over. Try it on your polynomial with the Gaussian prime factor $2+i$.