I have tried doing this by splitting fields and then using the tower law but I have not had any luck. Any help is much appreciated.
Question: Find $[\mathbb{Q}({5}^{1/4}):\mathbb{Q}]$ and the minimal polynomial of a over $\mathbb{Q}$.
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It is $f(x)=x^4-5$, and the degree is $4$, see this question. – Dietrich Burde Mar 09 '18 at 19:43
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Presumably, $a$ is $5^{1/4}?$ – Thomas Andrews Mar 09 '18 at 19:51
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Is it possible to show this by splitting field and tower law? As I do not understand the question linked. Thanks – Boj Mar 09 '18 at 20:00
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Presumably you want to exhibit the extension$\Bbb Q(5^{1/4})\supset\Bbb Q$ as a tower of two quadratic extensions. The only intermediate extension is $\Bbb Q(\sqrt2,)$. – Lubin Mar 09 '18 at 21:58
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$5^{1/4}$ satisfies $x^4-5$ which is irreducible over $\mathbb{Q}$ by Eisenstein's criterion. So it is the minimal polynomial of $5^{1/4}$ and $[\mathbb{Q}(5^{1/4}):\mathbb{Q}]=\deg{x^4-5}=4$. So the degree of the extension is $4$.
Jonathan Dunay
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