Factorize $x^4-4$ in $\mathbb{Q}[x]/(x^4+1)[x]$

Question

What are the irreducible factors of $x^4-4$ in $K[x]$ where $K=\mathbb Q[x]/(x^4+1)$?

Clearly $x^4-4=(x^2+2)(x^2-2)$, but from here I'm not sure what to do. I tried showing that $\sqrt{2}\in K$ or $\sqrt{2}\not\in K$ but couldn't do either.

Answer 1

$K=\mathbb Q[x]/(x^4+1)=\mathbb Q(a)$, where $a^4+1=0$. This shows that $a$ is a primitive $8$th root of unity, so we may take $a=\frac{1}{\sqrt 2}(1+i)$. Show that $K=\mathbb Q(\sqrt 2,i)$. Now finish the proof.