Prove that $ x^4 - 2 $ is irreducible over $ \mathbb{Z}[i] $ 
How do I prove that $ p(x) = x^4 - 2 $ is irreducible over $ \mathbb{Z}[i] $? 

This seems very elementary yet I'm not sure how to do it.
Someone suggested using Eisenstein and $ p = 1+i $, but this doesn't seem right because $ (1+i)^2 = 2i $ is an associate of $ -2 $.
I have seen somewhere that one can use a generalized version of the Rational Root Theorem and simply check that $ 1+i $ and $ 1-i $ are not roots of $ p(x) $, is this correct?
Thank you for your help.

You can also note that $x^4-2$ is irreducible over $\mathbb F_5$ and $\mathbb F_5 = \mathbb Z[i]/(2+i)$.
