Questions about the splitting field $K$ of $(x^4-1)(x^4+2)$ over $\mathbb Q$ 
Find the splitting field $K$ of $(x^4-1)(x^4+2)$ over $\mathbb Q$. 
  What is degree of extension? 
  Find an element $\alpha \in K$ such that $K=\mathbb Q(\alpha)$. 

How do I find the field $K$ and the element $\alpha$?
My attempt:  In $\mathbb{Q}(i,2^{1/4})$ the polynomial splits, but is this the splitting field?  Also, can I say $\alpha=i+2^{1/4}$?

Since The splitting field of $(x^4-1)(x^4+2)$ is the same than the splitting field of $(x^2+1)(x^4+2)$ (why ?), it has degree 8. Moreover $\mathbb Q(i,2^{1/4})$ has degree $8$ and split $(x^4-1)(x^4+2)$. The claim follow.
