Find $[\mathbb{Q}({5}^{1/4}):\mathbb{Q}]$ and the minimal polynomial of a over Q.

Question

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 $\mathbb{Q}$.

Answer 1

$5^{1/4}$ 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 $4$.