Given $a=\sqrt{5}+\sqrt{-5}$ and $b=5^{\frac{1}{4}}$, find de degree of the field extension $\Bbb{Q}(a,b)\mid\Bbb{Q}(b)$
I tried to compute the minimal polynomials of $a$ and $b$, and I got
$$P_{\Bbb{Q},a}(t)=t^4+100$$ and
$$P_{\Bbb{Q},b}(t)=t^4-5$$
But both polynomials had degree four, which are not prime twins, so I can't conclude anything about de degree of $\Bbb{Q}(a,b)\mid\Bbb{Q}(b)$.
I also noticed that we can write $a$ as
$$a=b^2+ib^2$$ but I don't know how that would help to solve this problem. I'm looking for tips to point me to the right direction, not just an entire answer to this problem, but anyway, any help would be highly appreciate. Thanks in advance!
Based on your observation that $a = b^2 + ib^2$, we have that
$$\mathbb{Q}(a, b) = \mathbb{Q}(\sqrt[4]{5}, i)$$
You now want to compute the degree of $\mathbb{Q}(\sqrt[4]{5}, i)$ over $\mathbb{Q}(\sqrt[4]{5})$.
To this end, it suffices to figure out what the minimum polynomial of $i$ is over the small field.