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I have the four roots but I am unsure how to proceed next, and how to show the degree of the extension over Q is 8.

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Do you know what "splitting field" means? – Chris Eagle Oct 23 '12 at 20:12
it's the smallest field extension in which the polynomial splits, right? I have the 4 roots (via factoring), but not sure what to do next – Mark S Oct 23 '12 at 20:32
Hint: $(x^4+13)(x^4-13) = x^8 -169$. So a root of your polynomial is also... – Derek Allums Oct 23 '12 at 20:40

By looking at the roots $$\sqrt[4]{-13},i\sqrt[4]{-13},-\sqrt[4]{-13},-i\sqrt[4]{-13}$$ it is clear that the splitting field is $$\mathbb{Q}(i,\sqrt[4]{-13})$$ that is since both $i\sqrt[4]{-13},\sqrt[4]{-13}$ is in the splitting field hence $\frac{i\sqrt[4]{-13}}{\sqrt[4]{-13}}=i$ is.

Now we need to find $[\mathbb{Q}(i,\sqrt[4]{-13}):\mathbb{Q}]$.

Argue that $i\not\in\mathbb{Q}(\sqrt[4]{-13})$ by using that fact that $\sqrt{13}\not\in\mathbb{Q}$, deduce that the degree is $8$ by noting $[\mathbb{Q}(\sqrt[4]{-13}):\mathbb{Q}]=4$ since it is a simple extension and $x^{4}+13\in\mathbb{Q}[x]$ is irreducible

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