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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 $\alpha=i+2^{1/4}$?

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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.

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