Extension field over the field of rational numbers

Question

$[\mathbb{Q}(2^{1/4}+8^{1/2}):\mathbb{Q}]=\text{degree(minimal polynomial)}$

I think that the minimal polynomial is $(x^4+48x^2+62)^2=2(8x^3-64x)^2$ with degree $ 8$, but it's not irreducible by Eisenstein's theorem.

any suggest, for resolution?

Answer 1

Since $\sqrt{8}=2\sqrt{2}$, once you add $\sqrt[4]{2}$, you automatically added its square ($\sqrt{2}$) and so also added its square doubled ($2\sqrt{2}$). Hence you are only adding $\sqrt[4]{2}$.