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.
how would I find the splitting field of $x^4+13$?
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field-theory
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0Hint: $(x^4+13)(x^4-13) = x^8 -169$. So a root of your polynomial is also... – 2012-10-23
1 Answers
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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