I was reading the lecture notes of Prof. Conrad, namely example 5 on the page 7 and ran into the following unclear moment on the page 9:
Question: He want to show that $[\mathbb{Q(\alpha+\frac{i}{\alpha}):\mathbb{Q}}]=4$. Suppose it is not $4$ then he claims that this field is inside $\mathbb{Q}(i)$.
Honestly to say yesterday I've spens alomost a whole day trying to derive it but failed to do it.
Can anyone show how did he get this ?
Would be very grateful for help!
EDIT: I can show directly that $[\mathbb{Q}(\alpha+\frac{i}{\alpha}):\mathbb{Q}]=4$ because $\alpha+\frac{i}{\alpha}$ is a root of polynomial $x^4-2x^2+5$ which is irreducible over rationals. But I would like to understand how the author is doing this. Namely why this extension is a subfield of $\mathbb{Q}(i)$?
